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Transformation of Functions

Graph functions using vertical and horizontal shifts.

Transformation of Functions

Identify graphs of basic functions, (IA 3.6.2)
Graph quadratic functions using transformations, (IA 9.7.4)

Basic functions have unique shapes, characteristics, and algebraic equations. It will be helpful to recognize and identify these basic or “toolkit functions” in our work in algebra, precalculus and calculus. Remember functions can be represented in many ways including by name, equation, graph, and basic tables of values.

Use a graphing program to help complete the following. Then, choose three values of x to evaluate for each. Add the x and y to the table for each exercise.

Try it.

Name	Equation	Graph
Constant	y=c, where c is a constant

Choose 3 values of x to evaluate for each.

x	y

Try it.

Name	Equation	Graph
Identity	y=x

Choose 3 values of x to evaluate for each.

x	y

Try it.

Name	Equation	Graph
Absolute Value	y=\|x\|

Choose 3 values of x to evaluate for each.

x	y

Try it.

Name	Equation	Graph
Quadratic	y=x²

Choose 3 values of x to evaluate for each.

x	y

Try it.

Name	Equation	Graph
Cubic	y=x³

Choose 3 values of x to evaluate for each.

x	y

Try it.

Name	Equation	Graph
Reciprocal	\(y=\frac{1}{x}\)

Choose 3 values of x to evaluate for each.

x	y

Try it.

Name	Equation	Graph
Square Root	\(y=\sqrt{x}\)

Choose 3 values of x to evaluate for each.

x	y

Try it.

Name	Equation	Graph
Cube Root	\(y=\sqrt[3]{x}\)

Choose 3 values of x to evaluate for each.

x	y

Try it.

Name	Equation	Graph
Exponential	\(y={e}^{x}\)

Choose 3 values of x to evaluate for each.

x	y

Condensed — the full section is in OpenStax College Algebra 2e.

Graphing Functions Using Vertical and Horizontal Shifts

Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.

Condensed — the full section is in OpenStax College Algebra 2e.

Graphing Functions Using Reflections about the Axes

Another transformation that can be applied to a function is a reflection over the x- or y-axis. A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis. The reflections are shown in .

Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the x-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the y-axis.

Example

Try it.

Reflect the graph of \(s(t)=\sqrt{t}\) (a) vertically and (b) horizontally.

Solution

ⓐ
Reflecting the graph vertically means that each output value will be reflected over the horizontal t-axis as shown in .

Because each output value is the opposite of the original output value, we can write
\[V(t)=-s(t)\ \text{or }V(t)=-\sqrt{t}\]
Notice that this is an outside change, or vertical shift, that affects the output \(s(t)\) values, so the negative sign belongs outside of the function.
ⓑ
Reflecting horizontally means that each input value will be reflected over the vertical axis as shown in .

Because each input value is the opposite of the original input value, we can write
\[H(t)=s(-t)\ \text{or }H(t)=\sqrt{-t}\]
Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.

Note that these transformations can affect the domain and range of the functions. While the original square root function has domain \([0,\infty )\) and range \([0,\infty ),\) the vertical reflection gives the \(V(t)\) function the range \((-\infty ,\ 0]\) and the horizontal reflection gives the \(H(t)\) function the domain \((-\infty ,\ 0].\)

Condensed — the full section is in OpenStax College Algebra 2e.

Determining Even and Odd Functions

Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions \(f(x)={x}^{2}\) or \(f(x)=|x|\) will result in the original graph. We say that these types of graphs are symmetric about the y-axis. A function whose graph is symmetric about the y-axis is called an even function.

If the graphs of \(f(x)={x}^{3}\) or \(f(x)=\frac{1}{x}\) were reflected over both axes, the result would be the original graph, as shown in .

We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an odd function.

Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, \(f(x)={2}^{x}\) is neither even nor odd. Also, the only function that is both even and odd is the constant function \(f(x)=0.\)

Example

Try it.

Is the function \(f(x)={x}^{3}+2x\) even, odd, or neither?

Solution

Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let’s begin with the rule for even functions.

\[f(-x)={(-x)}^{3}+2(-x)=-{x}^{3}-2x\]

This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.

\[-f(-x)=-(-{x}^{3}-2x)={x}^{3}+2x\]

Because \(-f(-x)=f(x),\) this is an odd function.

Graphing Functions Using Stretches and Compressions

Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.

We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.

Condensed — the full section is in OpenStax College Algebra 2e.

Performing a Sequence of Transformations

When combining transformations, it is very important to consider the order of the transformations. For example, vertically shifting by 3 and then vertically stretching by 2 does not create the same graph as vertically stretching by 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first.

When we see an expression such as \(2f(x)+3,\) which transformation should we start with? The answer here follows nicely from the order of operations. Given the output value of \(f(x),\) we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. In other words, multiplication before addition.

Horizontal transformations are a little trickier to think about. When we write \(g(x)=f(2x+3),\) for example, we have to think about how the inputs to the function \(g\) relate to the inputs to the function \(f.\) Suppose we know \(f(7)=12.\) What input to \(g\) would produce that output? In other words, what value of \(x\) will allow \(g(x)=f(2x+3)=12?\) We would need \(2x+3=7.\) To solve for \(x,\) we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression.

This format ends up being very difficult to work with, because it is usually much easier to horizontally stretch a graph before shifting. We can work around this by factoring inside the function.

\[f(bx+p)=f(b(x+\frac{p}{b}))\]

Let’s work through an example.

\[f(x)={(2x+4)}^{2}\]

We can factor out a 2.

\[f(x)={(2(x+2))}^{2}\]

Now we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. Factoring in this way allows us to horizontally stretch first and then shift horizontally.

Condensed — the full section is in OpenStax College Algebra 2e.

Key Equations

Vertical shift	\(g(x)=f(x)+k\) (up for \(k>0\) )
Horizontal shift	\(g(x)=f(x-h)\) (right for \(h>0\) )
Vertical reflection	\(g(x)=-f(x)\)
Horizontal reflection	\(g(x)=f(-x)\)
Vertical stretch	\(g(x)=af(x)\) ( \(a>0\) )
Vertical compression	\(g(x)=af(x)\) \((0
Horizontal stretch	\(g(x)=f(bx)\) \((0
Horizontal compression.	\(g(x)=f(bx)\) ( \(b>1\) )

Key Concepts

A function can be shifted vertically by adding a constant to the output. See and .
A function can be shifted horizontally by adding a constant to the input. See , , and .
Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See .
Vertical and horizontal shifts are often combined. See and .
A vertical reflection reflects a graph about the \(x\text{-}\) axis. A graph can be reflected vertically by multiplying the output by –1.
A horizontal reflection reflects a graph about the \(y\text{-}\) axis. A graph can be reflected horizontally by multiplying the input by –1.
A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. See .
A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See .
A function presented as an equation can be reflected by applying transformations one at a time. See .
Even functions are symmetric about the \(y\text{-}\) axis, whereas odd functions are symmetric about the origin.
Even functions satisfy the condition \(f(x)=f(-x).\)
Odd functions satisfy the condition \(f(x)=-f(-x).\)
A function can be odd, even, or neither. See .
A function can be compressed or stretched vertically by multiplying the output by a constant. See , , and .
A function can be compressed or stretched horizontally by multiplying the input by a constant. See , , and .
The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order. See and .

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Name Equation Graph
Constant y=c, where c is a constant
Choose 3 values of x to evaluate for each.
x y
Name Equation Graph
Identity y=x
Choose 3 values of x to evaluate for each.
x y
Name Equation Graph
Absolute Value y=|x|
Choose 3 values of x to evaluate for each.
x y
Name Equation Graph
Quadratic y=x²
Choose 3 values of x to evaluate for each.
x y
Name Equation Graph
Cubic y=x³
Choose 3 values of x to evaluate for each.
x y
Name Equation Graph
Reciprocal \(y=\frac{1}{x}\)
Choose 3 values of x to evaluate for each.
x y
Name Equation Graph
Square Root \(y=\sqrt{x}\)
Choose 3 values of x to evaluate for each.
x y
Name Equation Graph
Cube Root \(y=\sqrt[3]{x}\)
Choose 3 values of x to evaluate for each.
x y
Name Equation Graph
Exponential \(y={e}^{x}\)
Choose 3 values of x to evaluate for each.
x y
In general, what effect does adding or subtracting a constant have on the graph of \(f(x)={x}^{2}\) ?

\(f(x)={x}^{2}\)
vertex: ________
\(f(x)={x}^{2}+2\)
vertex: ________
\(f(x)={x}^{2}-4\)
vertex: ________
In general, what effect does adding or subtracting a value to x before it is squared have on the graph of \(f(x)={x}^{2}\) ?

\(f(x)={(x-2)}^{2}\)
vertex: ________
\(f(x)={(x-4)}^{2}\)
vertex: ________
\(f(x)={(x+3)}^{2}\)
vertex: ________
In general, what effect does multiplying by a constant have on the graph of \(f(x)={x}^{2}\) ?

\(f(x)=2{x}^{2}\)
vertex: ________
\(f(x)=5{x}^{2}\)
vertex: ________
\(f(x)=\frac{1}{10}{x}^{2}\)
vertex: ________
In general, what effect does multiplying by a negative constant have on the graph of \(f(x)={x}^{2}\) ?

\(f(x)=-{x}^{2}\)
vertex: ________
\(f(x)=-3{x}^{2}\)
vertex: ________
\(y={(x-2)}^{2}\)
vertex: ________
\(f(x)={(x+3)}^{2}+2\)
vertex: ________
\(f(x)={(x+3)}^{2}-2\)
vertex: ________
\(f(x)={(x-3)}^{2}+2\)
vertex: ________
\(f(x)={(x-3)}^{2}-2\)
vertex: ________
Answer each of the following based on the changes you saw in the graphs above.
ⓐ Based on your observations from the previous graphs, what are the coordinates of the vertex of the parabola \(f(x)={(x+200)}^{2}-67\) ? Do not attempt to graph!
ⓑ Based on your observations from the previous graphs, what are the coordinates of the vertex of the parabola \(f(x)=12{(x+6)}^{2}+111\) ? Do not attempt to graph!
Fill in the blanks:
ⓐ If c > 0, the graph of \(y=f(x)+c\) is obtained by shifting the graph of \(y=f(x)\) to the ________ a distance of c units. The graph of \(y=f(x)-c\) is obtained by shifting the graph of \(y=f(x)\) to the ________ a distance of c units.
ⓑ If c > 0, the graph of \(y=f(x)-c\) is obtained by shifting the graph of \(y=f(x)\) to the ________ a distance of c units. The graph of \(y=f(x)+c\) is obtained by shifting the graph of \(y=f(x)\) to the ________ a distance of c units.
Apply what you have learned in this skill sheet regarding transformations. Write the equation of a quadratic function that has been transformed in the each of the ways described in parts ⓐ and ⓑ below. Write equations in f(x) = form. After writing an equation check your answer using a graphing program and graph below. Be sure to label the vertex as an ordered pair. Does your graph match the description?
ⓐ Flipped upside down and shifted to the right 3 units and down 2 units.
f(x) =

ⓑ Stretched vertically by a factor of 4 and shifted left 6 units and up 5 units.
f(x) =
Remember the basic transformations investigated in this activity apply to all basic functions. Apply what you have learned in this lab about transformations. Write the equation of a function that has been transformed in the following ways. Write equations in f(x) = form. After writing an equation check your answer using a graphing program and graph below. Be sure to label a point on the graph. Does your graph match the description?
ⓐ Begin with a basic square root function. Reflect the graph over the x-axis and shift it to the right 2 units and down 1 unit.
f(x) =

ⓑ Begin with an absolute value function. Stretch the graph vertically by a factor of 3 and shift it left 4 units and up 5 units.
f(x) =
To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. shows the area of open vents \(V\) (in square feet) throughout the day in hours after midnight, \(t.\) During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.

جواب رو نشون بده

We can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will have the effect of shifting the graph vertically up, as shown in .
Notice that in , for each input value, the output value has increased by 20, so if we call the new function \(S(t),\) we could write
\[S(t)=V(t)+20\]
This notation tells us that, for any value of \(t,S(t)\) can be found by evaluating the function \(V\) at the same input and then adding 20 to the result. This defines \(S\) as a transformation of the function \(V,\) in this case a vertical shift up 20 units. Notice that, with a vertical shift, the input values stay the same and only the output values change. See .
\(t\) 0 8 10 17 19 24
\(V(t)\) 0 0 220 220 0 0
\(S(t)\) 20 20 240 240 20 20
A function \(f(x)\) is given in . Create a table for the function \(g(x)=f(x)-3.\)
\(x\) 2 4 6 8
\(f(x)\) 1 3 7 11

جواب رو نشون بده

The formula \(g(x)=f(x)-3\) tells us that we can find the output values of \(g\) by subtracting 3 from the output values of \(f.\) For example:
\[\begin{array}{llll}f(2) & = & 1 & \text{Given} \\ g(x) & = & f(x)-3 & \text{Given transformation} \\ g(2) & = & f(2)-3 & \\ & = & 1-3 & \\ & = & -2 & \end{array}\]
Subtracting 3 from each \(f(x)\) value, we can complete a table of values for \(g(x)\) as shown in .
\(x\) 2 4 6 8
\(f(x)\) 1 3 7 11
\(g(x)\) −2 0 4 8
The function \(h(t)=-4.9{t}^{2}+30t\) gives the height \(h\) of a ball (in meters) thrown upward from the ground after \(t\) seconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height function \(b(t)\) to \(h(t),\) and then find a formula for \(b(t).\)

جواب رو نشون بده

\[b(t)=h(t)+10=-4.9{t}^{2}+30t+10\]
Returning to our building airflow example from , suppose that in autumn the facilities manager decides that the original venting plan starts too late, and wants to begin the entire venting program 2 hours earlier. Sketch a graph of the new function.

جواب رو نشون بده

We can set \(V(t)\) to be the original program and \(F(t)\) to be the revised program.
\[\begin{array}{lll}V(t) & = & \text{the original venting plan} \\ F(t) & = & \text{starting 2 hrs sooner}\end{array}\]
In the new graph, at each time, the airflow is the same as the original function \(V\) was 2 hours later. For example, in the original function \(V,\) the airflow starts to change at 8 a.m., whereas for the function \(F,\) the airflow starts to change at 6 a.m. The comparable function values are \(V(8)=F(6).\) See . Notice also that the vents first opened to \(220{\ \text{ft}}^{2}\) at 10 a.m. under the original plan, while under the new plan the vents reach \(220{\ \text{ft}}^{\text{2}}\) at
8 a.m., so \(V(10)=F(8).\)
In both cases, we see that, because \(F(t)\) starts 2 hours sooner, \(h=-2.\) That means that the same output values are reached when \(F(t)=V(t-(-2))=V(t+2).\)
A function \(f(x)\) is given in . Create a table for the function \(g(x)=f(x-3).\)
\(x\) 2 4 6 8
\(f(x)\) 1 3 7 11

جواب رو نشون بده

The formula \(g(x)=f(x-3)\) tells us that the output values of \(g\) are the same as the output value of \(f\) when the input value is 3 less than the original value. For example, we know that \(f(2)=1.\) To get the same output from the function \(g,\) we will need an input value that is 3 larger. We input a value that is 3 larger for \(g(x)\) because the function takes 3 away before evaluating the function \(f.\)
\[\begin{array}{lll}g(5) & = & f(5-3) \\ & = & f(2) \\ & = & 1\end{array}\]
We continue with the other values to create .
\(x\) 5 7 9 11
\(x-3\) 2 4 6 8
\(f(x-3)\) 1 3 7 11
\(g(x)\) 1 3 7 11
The result is that the function \(g(x)\) has been shifted to the right by 3. Notice the output values for \(g(x)\) remain the same as the output values for \(f(x),\) but the corresponding input values, \(x,\) have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.
represents a transformation of the toolkit function \(f(x)={x}^{2}.\) Relate this new function \(g(x)\) to \(f(x),\) and then find a formula for \(g(x).\)

جواب رو نشون بده

Notice that the graph is identical in shape to the \(f(x)={x}^{2}\) function, but the x-values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so
\[g(x)=f(x-2)\]
Notice how we must input the value \(x=2\) to get the output value \(y=0;\) the x-values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the \(f(x)\) function to write a formula for \(g(x)\) by evaluating \(f(x-2).\)
\[\begin{array}{lll}f(x) & = & {x}^{2} \\ g(x) & = & f(x-2) \\ g(x) & = & f(x-2)={(x-2)}^{2}\end{array}\]
The function \(G(m)\) gives the number of gallons of gas required to drive \(m\) miles. Interpret \(G(m)+10\) and \(G(m+10).\)

جواب رو نشون بده

\(G(m)+10\) can be interpreted as adding 10 to the output, gallons. This is the gas required to drive \(m\) miles, plus another 10 gallons of gas. The graph would indicate a vertical shift.
\(G(m+10)\) can be interpreted as adding 10 to the input, miles. So this is the number of gallons of gas required to drive 10 miles more than \(m\) miles. The graph would indicate a horizontal shift.
Given the function \(f(x)=\sqrt{x},\) graph the original function \(f(x)\) and the transformation \(g(x)=f(x+2)\) on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?

جواب رو نشون بده

The graphs of \(f(x)\) and \(g(x)\) are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.
Given \(f(x)=|x|,\) sketch a graph of \(h(x)=f(x+1)-3.\)
جواب رو نشون بده
The function \(f\) is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of \(h\) has transformed \(f\) in two ways: \(f(x+1)\) is a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 in \(f(x+1)-3\) is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated in .
Let us follow one point of the graph of \(f(x)=|x|.\)
- The point \((0,0)\) is transformed first by shifting left 1 unit: \((0,0)\to (-1,0)\)
- The point \((-1,0)\) is transformed next by shifting down 3 units: \((-1,0)\to (-1,-3)\)
shows the graph of \(h.\)
Given \(f(x)=|x|,\) sketch a graph of \(h(x)=f(x-2)+4.\)
Write a formula for the graph shown in , which is a transformation of the toolkit square root function.

جواب رو نشون بده

The graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function notation, we could write that as
\[h(x)=f(x-1)+2\]
Using the formula for the square root function, we can write
\[h(x)=\sqrt{x-1}+2\]
Write a formula for a transformation of the toolkit reciprocal function \(f(x)=\frac{1}{x}\) that shifts the function’s graph one unit to the right and one unit up.

جواب رو نشون بده

\(g(x)=\frac{1}{x-1}+1\)
Reflect the graph of \(s(t)=\sqrt{t}\) (a) vertically and (b) horizontally.
جواب رو نشون بده
1. ⓐ
  Reflecting the graph vertically means that each output value will be reflected over the horizontal t-axis as shown in .
  
  Because each output value is the opposite of the original output value, we can write
  \[V(t)=-s(t)\ \text{or }V(t)=-\sqrt{t}\]
  Notice that this is an outside change, or vertical shift, that affects the output \(s(t)\) values, so the negative sign belongs outside of the function.
2. ⓑ
  Reflecting horizontally means that each input value will be reflected over the vertical axis as shown in .
  
  Because each input value is the opposite of the original input value, we can write
  \[H(t)=s(-t)\ \text{or }H(t)=\sqrt{-t}\]
  Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.
  
  Note that these transformations can affect the domain and range of the functions. While the original square root function has domain \([0,\infty )\) and range \([0,\infty ),\) the vertical reflection gives the \(V(t)\) function the range \((-\infty ,\ 0]\) and the horizontal reflection gives the \(H(t)\) function the domain \((-\infty ,\ 0].\)
Reflect the graph of \(f(x)=|x-1|\) (a) vertically and (b) horizontally.
جواب رو نشون بده
1. ⓐ
2. ⓑ
A function \(f(x)\) is given as . Create a table for the functions below.
1. ⓐ \(g(x)=-f(x)\)
2. ⓑ \(h(x)=f(-x)\)
\(x\) 2 4 6 8
\(f(x)\) 1 3 7 11
جواب رو نشون بده
1. ⓐ
  For \(g(x),\) the negative sign outside the function indicates a vertical reflection, so the x-values stay the same and each output value will be the opposite of the original output value. See .
  \(x\) 2 4 6 8
  \(g(x)\) –1 –3 –7 –11
2. ⓑ
  For \(h(x),\) the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the \(h(x)\) values stay the same as the \(f(x)\) values. See .
  \(x\) −2 −4 −6 −8
  \(h(x)\) 1 3 7 11
A function \(f(x)\) is given as . Create a table for the functions below.
1. ⓐ \(g(x)=-f(x)\)
2. ⓑ \(h(x)=f(-x)\)
\(x\) −2 0 2 4
\(f(x)\) 5 10 15 20
جواب رو نشون بده
1. ⓐ
  \(g(x)=-f(x)\)
  
  \(x\) -2 0 2 4
  \(g(x)\) \(-5\) \(-10\) \(-15\) \(-20\)
2. ⓑ
  \(h(x)=f(\text{-}x)\)
  \(x\) -2 0 2 4
  \(h(x)\) 15 10 5 unknown
A common model for learning has an equation similar to \(k(t)=-{2}^{-t}+1,\) where \(k\) is the percentage of mastery that can be achieved after \(t\) practice sessions. This is a transformation of the function \(f(t)={2}^{t}\) shown in . Sketch a graph of \(k(t).\)
جواب رو نشون بده
This equation combines three transformations into one equation.
- A horizontal reflection: \(f(\text{-}t)={2}^{-t}\)
- A vertical reflection: \(-f(\text{-}t)=-{2}^{-t}\)
- A vertical shift: \(-f(\text{-}t)+1=-{2}^{-t}+1\)
We can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points (0, 1) and (1, 2).
1. First, we apply a horizontal reflection: (0, 1) (–1, 2).
2. Then, we apply a vertical reflection: (0, -1) (-1, –2)
3. Finally, we apply a vertical shift: (0, 0) (-1, -1)).
This means that the original points, (0,1) and (1,2) become (0,0) and (-1,-1) after we apply the transformations.
In , the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.
Given the toolkit function \(f(x)={x}^{2},\) graph \(g(x)=-f(x)\) and \(h(x)=f(-x).\) Take note of any surprising behavior for these functions.

جواب رو نشون بده

Notice: \(g(x)=f(-x)\) looks the same as \(f(x)\) .
Is the function \(f(x)={x}^{3}+2x\) even, odd, or neither?

جواب رو نشون بده

Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let’s begin with the rule for even functions.
\[f(-x)={(-x)}^{3}+2(-x)=-{x}^{3}-2x\]
This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.
\[-f(-x)=-(-{x}^{3}-2x)={x}^{3}+2x\]
Because \(-f(-x)=f(x),\) this is an odd function.
Is the function \(f(s)={s}^{4}+3{s}^{2}+7\) even, odd, or neither?

جواب رو نشون بده

even
A function \(P(t)\) models the population of fruit flies. The graph is shown in .
A scientist is comparing this population to another population, \(Q,\) whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.

جواب رو نشون بده

Because the population is always twice as large, the new population’s output values are always twice the original function’s output values. Graphically, this is shown in .
If we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2.
The following shows where the new points for the new graph will be located.
\[\begin{array}{l}(0,\ 1)\to (0,\ 2) \\ (3,\ 3)\to (3,\ 6) \\ (6,\ 2)\to (6,\ 4) \\ (7,\ 0)\to (7,\ 0)\end{array}\]
Symbolically, the relationship is written as
\[Q(t)=2P(t)\]
This means that for any input \(t,\) the value of the function \(Q\) is twice the value of the function \(P.\) Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values, \(t,\) stay the same while the output values are twice as large as before.
A function \(f\) is given as . Create a table for the function \(g(x)=\frac{1}{2}f(x).\)
\(x\) 2 4 6 8
\(f(x)\) 1 3 7 11

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The formula \(g(x)=\frac{1}{2}f(x)\) tells us that the output values of \(g\) are half of the output values of \(f\) with the same inputs. For example, we know that \(f(4)=3.\) Then
\[g(4)=\frac{1}{2}f(4)=\frac{1}{2}(3)=\frac{3}{2}\]
We do the same for the other values to produce .
\(x\) \(2\) \(4\) \(6\) \(8\)
\(g(x)\) \(\frac{1}{2}\) \(\frac{3}{2}\) \(\frac{7}{2}\) \(\frac{11}{2}\)

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

i

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Transformation of Functions

Graph functions using vertical and horizontal shifts.
Graph functions using reflections about the x-axis and the y-axis.
Determine whether a function is even, odd, or neither from its graph.
Graph functions using compressions and stretches.
Combine transformations.
Identify graphs of basic functions, (IA 3.6.2)
Graph quadratic functions using transformations, (IA 9.7.4)
Identify the output row or column.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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Parts of this page are adapted from OpenStax College Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

بیشتر در Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value