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Graph Quadratic Functions Using Transformations

Graph quadratic equations of the form

Graph Quadratic Functions of the form

In the last section, we learned how to graph quadratic functions using their properties. Another method involves starting with the basic graph of \(f(x)={x}^{2}\) and ‘moving’ it according to information given in the function equation. We call this graphing quadratic functions using transformations.

In the first example, we will graph the quadratic function \(f(x)={x}^{2}\) by plotting points. Then we will see what effect adding a constant, k, to the equation will have on the graph of the new function \(f(x)={x}^{2}+k.\)

Example

Try it.

Graph \(f(x)={x}^{2},\ g(x)={x}^{2}+2,\) and \(h(x)={x}^{2}-2\) on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.

Solution

Plotting points will help us see the effect of the constants on the basic \(f(x)={x}^{2}\) graph. We fill in the chart for all three functions.

The g(x) values are two more than the f(x) values. Also, the h(x) values are two less than the f(x) values. Now we will graph all three functions on the same rectangular coordinate system.

The graph of \(g(x)={x}^{2}+2\) is the same as the graph of \(f(x)={x}^{2}\) but shifted up 2 units.

The graph of \(h(x)={x}^{2}-2\) is the same as the graph of \(f(x)={x}^{2}\) but shifted down 2 units.

The last example shows us that to graph a quadratic function of the form \(f(x)={x}^{2}+k,\) we take the basic parabola graph of \(f(x)={x}^{2}\) and vertically shift it up \((k>0)\) or shift it down \((k<0)\).

This transformation is called a vertical shift.

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

Graph Quadratic Functions of the form

In the first example, we graphed the quadratic function \(f(x)={x}^{2}\) by plotting points and then saw the effect of adding a constant k to the function had on the resulting graph of the new function \(f(x)={x}^{2}+k.\)

We will now explore the effect of subtracting a constant, h, from x has on the resulting graph of the new function \(f(x)={(x-h)}^{2}.\)

Example

Try it.

Graph \(f(x)={x}^{2},\ g(x)={(x-1)}^{2},\) and \(h(x)={(x+1)}^{2}\) on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.

Solution

Plotting points will help us see the effect of the constants on the basic \(f(x)={x}^{2}\) graph. We fill in the chart for all three functions.

The g(x) values and the h(x) values share the common numbers 0, 1, 4, 9, and 16, but are shifted.

The last example shows us that to graph a quadratic function of the form \(f(x)={(x-h)}^{2},\) we take the basic parabola graph of \(f(x)={x}^{2}\) and shift it left (h > 0) or shift it right (h < 0).

This transformation is called a horizontal shift.

Now that we have seen the effect of the constant, h, it is easy to graph functions of the form \(f(x)={(x-h)}^{2}.\) We just start with the basic parabola of \(f(x)={x}^{2}\) and then shift it left or right.

The next example will require a horizontal shift.

Example

Try it.

Graph \(f(x)={(x-5)}^{2}\) using a horizontal shift.

Solution

We first draw the graph of \(f(x)={x}^{2}\) on the grid.
Determine h.

Shift the graph \(f(x)={x}^{2}\) to the right 5 units.

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

Graph Quadratic Functions of the Form

So far we graphed the quadratic function \(f(x)={x}^{2}\) and then saw the effect of including a constant h or k in the equation had on the resulting graph of the new function. We will now explore the effect of the coefficient a on the resulting graph of the new function \(f(x)=a{x}^{2}.\)

If we graph these functions, we can see the effect of the constant a, assuming a > 0.

To graph a function with constant a it is easiest to choose a few points on \(f(x)={x}^{2}\) and multiply the y-values by a.

Example

Try it.

Graph \(f(x)=3{x}^{2}.\)

Solution

We will graph the functions \(f(x)={x}^{2}\) and \(g(x)=3{x}^{2}\) on the same grid. We will choose a few points on \(f(x)={x}^{2}\) and then multiply the y-values by 3 to get the points for \(g(x)=3{x}^{2}.\)

Graph Quadratic Functions Using Transformations

We have learned how the constants a, h, and k in the functions, \(f(x)={x}^{2}+k,\ f(x)={(x-h)}^{2},\) and \(f(x)=a{x}^{2}\) affect their graphs. We can now put this together and graph quadratic functions \(f(x)=a{x}^{2}+bx+c\) by first putting them into the form \(f(x)=a{(x-h)}^{2}+k\) by completing the square. This form is sometimes known as the vertex form or standard form.

We must be careful to both add and subtract the number to the SAME side of the function to complete the square. We cannot add the number to both sides as we did when we completed the square with quadratic equations.

When we complete the square in a function with a coefficient of x² that is not one, we have to factor that coefficient from just the x-terms. We do not factor it from the constant term. It is often helpful to move the constant term a bit to the right to make it easier to focus only on the x-terms.

Once we get the constant we want to complete the square, we must remember to multiply it by that coefficient before we then subtract it.

Example

Try it.

Rewrite \(f(x)=-3{x}^{2}-6x-1\) in the \(f(x)=a{(x-h)}^{2}+k\) form by completing the square.

Solution


Separate the x terms from the constant.
Factor the coefficient of \({x}^{2}\), \(-3\).
Prepare to complete the square.
Take half of 2 and then square it to complete the square. \({(\frac{1}{2}\cdot 2)}^{2}=1\)
The constant 1 completes the square in the parentheses, but the parentheses is multiplied by \(-3\). So we are really adding \(-3\) We must then add 3 to not change the value of the function.
Rewrite the trinomial as a square and subtract the constants.
The function is now in the \(f(x)=a{(x-h)}^{2}+k\) form.

Once we put the function into the \(f(x)={(x-h)}^{2}+k\) form, we can then use the transformations as we did in the last few problems. The next example will show us how to do this.

We list the steps to take to graph a quadratic function using transformations here.

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

Find a Quadratic Function from its Graph

So far we have started with a function and then found its graph.

Now we are going to reverse the process. Starting with the graph, we will find the function.

Example

Try it.

Determine the quadratic function whose graph is shown.

Solution

Since it is quadratic, we start with the \(f(x)=a{(x-h)}^{2}+k\ \text{form.}\)
The vertex, \((h,k),\) is \((-2,-1)\) so \(h=-2\) and \(k=-1.\)	\(\ f(x)=a{(x-(-2))}^{2}-1\)
To find \(a\), we use the \(y\)-intercept, \((0,7)\).
So \(f(0)=7\).	\(\ 7=a{(0+2)}^{2}-1\)
Solve for \(a\).	\(\ 7=4a-1\)
	\(\ 8=4a\)
	\(\ 2=a\)
Write the function.	\(\ f(x)=a{(x-h)}^{2}+k\)
Substitute in \(h=-2,k=-1\) and \(a=2\).	\(\ f(x)=2{(x+2)}^{2}-1\)

Key Concepts

Graph a Quadratic Function of the form \(f(x)={x}^{2}+k\) Using a Vertical Shift
- The graph of \(f(x)={x}^{2}+k\) shifts the graph of \(f(x)={x}^{2}\) vertically k units.
  - If k > 0, shift the parabola vertically up k units.
  - If k < 0, shift the parabola vertically down \(|k|\) units.
Graph a Quadratic Function of the form \(f(x)={(x-h)}^{2}\) Using a Horizontal Shift
- The graph of \(f(x)={(x-h)}^{2}\) shifts the graph of \(f(x)={x}^{2}\) horizontally h units.
  - If h > 0, shift the parabola horizontally left h units.
  - If h < 0, shift the parabola horizontally right \(|h|\) units.
Graph of a Quadratic Function of the form \(f(x)=a{x}^{2}\)
- The coefficient a in the function \(f(x)=a{x}^{2}\) affects the graph of \(f(x)={x}^{2}\) by stretching or compressing it.
  If \(0<|a|<1,\) then the graph of \(f(x)=a{x}^{2}\) will be “wider” than the graph of \(f(x)={x}^{2}.\)
  If \(|a|>1,\) then the graph of \(f(x)=a{x}^{2}\) will be “skinnier” than the graph of \(f(x)={x}^{2}.\)
How to graph a quadratic function using transformations
1. Rewrite the function in \(f(x)=a{(x-h)}^{2}+k\) form by completing the square.
2. Graph the function using transformations.
Graph a quadratic function in the vertex form \(f(x)=a{(x-h)}^{2}+k\) using properties
1. Rewrite the function in \(f(x)=a{(x-h)}^{2}+k\) form.
2. Determine whether the parabola opens upward, a > 0, or downward, a < 0.
3. Find the axis of symmetry, x = h.
4. Find the vertex, (h, k).
5. Find they-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
6. Find the x-intercepts, if possible.
7. Graph the parabola.

Graph Quadratic Functions Using Transformations

Graph Quadratic Functions of the form \(f(x)={x}^{2}+k\)

In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, k, to the function has on the basic parabola.

Try it.

\(f(x)={x}^{2},\ g(x)={x}^{2}+4,\) and \(h(x)={x}^{2}-4.\)

Solution

ⓐ

ⓑ The graph of \(g(x)={x}^{2}+4\) is the same as the graph of \(f(x)={x}^{2}\) but shifted up 4 units. The graph of \(h(x)={x}^{2}-4\) is the same as the graph of \(f(x)={x}^{2}\) but shift down 4 units.

Try it.

\(f(x)={x}^{2},\ g(x)={x}^{2}+7,\) and \(h(x)={x}^{2}-7.\)

In the following exercises, graph each function using a vertical shift.

Try it.

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

Solution

Try it.

\(f(x)={x}^{2}-7\)

Try it.

\(g(x)={x}^{2}+2\)

Solution

Try it.

\(g(x)={x}^{2}+5\)

Try it.

\(h(x)={x}^{2}-4\)

Solution

Try it.

\(h(x)={x}^{2}-5\)

Graph Quadratic Functions of the form \(f(x)={(x-h)}^{2}\)

In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, \(h\), inside the parentheses has

Try it.

\(f(x)={x}^{2},\ g(x)={(x-3)}^{2},\) and \(h(x)={(x+3)}^{2}.\)

Solution

ⓐ

ⓑ The graph of \(g(x)={(x-3)}^{2}\) is the same as the graph of \(f(x)={x}^{2}\) but shifted right 3 units. The graph of \(h(x)={(x+3)}^{2}\) is the same as the graph of \(f(x)={x}^{2}\) but shifted left 3 units.

Try it.

\(f(x)={x}^{2},\ g(x)={(x+4)}^{2},\) and \(h(x)={(x-4)}^{2}.\)

In the following exercises, graph each function using a horizontal shift.

Try it.

\(f(x)={(x-2)}^{2}\)

Solution

Try it.

\(f(x)={(x-1)}^{2}\)

Try it.

\(f(x)={(x+5)}^{2}\)

Solution

Try it.

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

Try it.

\(f(x)={(x-5)}^{2}\)

Solution

Try it.

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

In the following exercises, graph each function using transformations.

Try it.

\(f(x)={(x+2)}^{2}+1\)

Solution

Try it.

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

Try it.

\(f(x)={(x-1)}^{2}+5\)

Solution

Try it.

\(f(x)={(x-3)}^{2}+4\)

Try it.

\(f(x)={(x+3)}^{2}-1\)

Solution

Try it.

\(f(x)={(x+5)}^{2}-2\)

Try it.

\(f(x)={(x-4)}^{2}-3\)

Solution

Try it.

\(f(x)={(x-6)}^{2}-2\)

Graph Quadratic Functions of the form \(f(x)=a{x}^{2}\)

In the following exercises, graph each function.

Try it.

\(f(x)=-2{x}^{2}\)

Solution

Try it.

\(f(x)=4{x}^{2}\)

Try it.

\(f(x)=-4{x}^{2}\)

Solution

Try it.

\(f(x)=\text{-}{x}^{2}\)

Try it.

\(f(x)=\frac{1}{2}{x}^{2}\)

Solution

Try it.

\(f(x)=\frac{1}{3}{x}^{2}\)

Try it.

\(f(x)=\frac{1}{4}{x}^{2}\)

Solution

Try it.

\(f(x)=-\frac{1}{2}{x}^{2}\)

Graph Quadratic Functions Using Transformations

In the following exercises, rewrite each function in the \(f(x)=a{(x-h)}^{2}+k\) form by completing the square.

Try it.

\(f(x)=-3{x}^{2}-12x-5\)

Solution

\(f(x)=-3{(x+2)}^{2}+7\)

Try it.

\(f(x)=2{x}^{2}-12x+7\)

Try it.

\(f(x)=3{x}^{2}+6x-1\)

Solution

\(f(x)=3{(x+1)}^{2}-4\)

Try it.

\(f(x)=-4{x}^{2}-16x-9\)

In the following exercises, ⓐ rewrite each function in \(f(x)=a{(x-h)}^{2}+k\) form and ⓑ graph it by using transformations.

Try it.

\(f(x)={x}^{2}+6x+5\)

Solution

ⓐ \(f(x)={(x+3)}^{2}-4\)
ⓑ

Try it.

\(f(x)={x}^{2}+4x-12\)

Try it.

\(f(x)={x}^{2}+4x+3\)

Solution

ⓐ \(f(x)={(x+2)}^{2}-1\)
ⓑ

Try it.

\(f(x)={x}^{2}-6x+8\)

Try it.

\(f(x)={x}^{2}-6x+15\)

Solution

ⓐ \(f(x)={(x-3)}^{2}+6\)
ⓑ

Try it.

\(f(x)={x}^{2}+8x+10\)

Try it.

\(f(x)=\text{-}{x}^{2}+8x-16\)

Solution

ⓐ \(f(x)=\text{-}{(x-4)}^{2}+0\)
ⓑ

Try it.

\(f(x)=\text{-}{x}^{2}+2x-7\)

Try it.

\(f(x)=\text{-}{x}^{2}-4x+2\)

Solution

ⓐ \(f(x)=\text{-}{(x+2)}^{2}+6\)
ⓑ

Try it.

\(f(x)=\text{-}{x}^{2}+4x-5\)

Try it.

\(f(x)=5{x}^{2}-10x+8\)

Solution

ⓐ \(f(x)=5{(x-1)}^{2}+3\)
ⓑ

Try it.

\(f(x)=3{x}^{2}+18x+20\)

Try it.

\(f(x)=2{x}^{2}-4x+1\)

Solution

ⓐ \(f(x)=2{(x-1)}^{2}-1\)
ⓑ

Try it.

\(f(x)=3{x}^{2}-6x-1\)

Try it.

\(f(x)=-2{x}^{2}+8x-10\)

Solution

ⓐ \(f(x)=-2{(x-2)}^{2}-2\)
ⓑ

Try it.

\(f(x)=-3{x}^{2}+6x+1\)

In the following exercises, ⓐ rewrite each function in \(f(x)=a{(x-h)}^{2}+k\) form and ⓑ graph it using properties.

Try it.

\(f(x)=2{x}^{2}+4x+6\)

Solution

ⓐ \(f(x)=2{(x+1)}^{2}+4\)
ⓑ

Try it.

\(f(x)=3{x}^{2}-12x+7\)

Try it.

\(f(x)=\text{-}{x}^{2}+2x-4\)

Solution

ⓐ \(f(x)=\text{-}{(x-1)}^{2}-3\)
ⓑ

Try it.

\(f(x)=-2{x}^{2}-4x-5\)

Matching

In the following exercises, match the graphs to one of the following functions: ⓐ \(f(x)={x}^{2}+4\) ⓑ \(f(x)={x}^{2}-4\) ⓒ \(f(x)={(x+4)}^{2}\) ⓓ \(f(x)={(x-4)}^{2}\) ⓔ \(f(x)={(x+4)}^{2}-4\) ⓕ \(f(x)={(x+4)}^{2}+4\) ⓖ \(f(x)={(x-4)}^{2}-4\) ⓗ \(f(x)={(x-4)}^{2}+4\)

Try it.

Solution

ⓒ

Try it.

Solution

ⓔ

Try it.

Solution

ⓓ

Try it.

Solution

ⓖ

Try it.

Find a Quadratic Function from its Graph

In the following exercises, write the quadratic function in \(f(x)=a{(x-h)}^{2}+k\) form whose graph is shown.

Try it.

Solution

\(f(x)={(x+1)}^{2}-5\)

Try it.

Solution

\(f(x)=2{(x-1)}^{2}-3\)

Try it.

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

Practice (40)

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

Graph the function \(f(x)={x}^{2}\) by plotting points.
If you missed this problem, review .

Bonisa impendulo
Factor completely: \({y}^{2}-14y+49.\)
If you missed this problem, review .

Bonisa impendulo

\({\left({y}^{-7}\right)}^{2}\)
Factor completely: \(2{x}^{2}-16x+32.\)
If you missed this problem, review .

Bonisa impendulo

\(2{({x}^{-4})}^{2}\)
Graph \(f(x)={x}^{2},\ g(x)={x}^{2}+2,\) and \(h(x)={x}^{2}-2\) on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.

Bonisa impendulo

Plotting points will help us see the effect of the constants on the basic \(f(x)={x}^{2}\) graph. We fill in the chart for all three functions.
The g(x) values are two more than the f(x) values. Also, the h(x) values are two less than the f(x) values. Now we will graph all three functions on the same rectangular coordinate system.
The graph of \(g(x)={x}^{2}+2\) is the same as the graph of \(f(x)={x}^{2}\) but shifted up 2 units.
The graph of \(h(x)={x}^{2}-2\) is the same as the graph of \(f(x)={x}^{2}\) but shifted down 2 units.
ⓐ Graph \(f(x)={x}^{2},\ g(x)={x}^{2}+1,\) and \(h(x)={x}^{2}-1\) on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.

Bonisa impendulo

ⓐ

ⓑ The graph of \(g(x)={x}^{2}+1\) is the same as the graph of \(f(x)={x}^{2}\) but shifted up 1 unit. The graph of \(h(x)={x}^{2}-1\) is the same as the graph of \(f(x)={x}^{2}\) but shifted down 1 unit.
ⓐ Graph \(f(x)={x}^{2},\ g(x)={x}^{2}+6,\) and \(h(x)={x}^{2}-6\) on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.

Bonisa impendulo

ⓐ

ⓑ The graph of \(h(x)={x}^{2}+6\) is the same as the graph of \(f(x)={x}^{2}\) but shifted up 6 units. The graph of \(h(x)={x}^{2}-6\) is the same as the graph of \(f(x)={x}^{2}\) but shifted down 6 units.
Graph \(f(x)={x}^{2}-3\) using a vertical shift.

Bonisa impendulo

We first draw the graph of \(f(x)={x}^{2}\) on
the grid.
Determine \(k\).
Shift the graph \(f(x)={x}^{2}\) down 3.
Graph \(f(x)={x}^{2}-5\) using a vertical shift.

Bonisa impendulo
Graph \(f(x)={x}^{2}+7\) using a vertical shift.

Bonisa impendulo
Graph \(f(x)={x}^{2},\ g(x)={(x-1)}^{2},\) and \(h(x)={(x+1)}^{2}\) on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.

Bonisa impendulo

Plotting points will help us see the effect of the constants on the basic \(f(x)={x}^{2}\) graph. We fill in the chart for all three functions.

The g(x) values and the h(x) values share the common numbers 0, 1, 4, 9, and 16, but are shifted.
ⓐ Graph \(f(x)={x}^{2},\ g(x)={(x+2)}^{2},\) and \(h(x)={(x-2)}^{2}\) on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.

Bonisa impendulo

ⓐ

ⓑ The graph of \(g(x)={(x+2)}^{2}\) is the same as the graph of \(f(x)={x}^{2}\) but shifted left 2 units. The graph of \(h(x)={(x-2)}^{2}\) is the same as the graph of \(f(x)={x}^{2}\) but shift right 2 units.
ⓐ Graph \(f(x)={x}^{2},\ g(x)={x}^{2}+5,\) and \(h(x)={x}^{2}-5\) on the same rectangular coordinate system.
ⓑ Describe what effect adding a constant to the function has on the basic parabola.

Bonisa impendulo

ⓐ

ⓑ The graph of \(g(x)={(x+5)}^{2}\) is the same as the graph of \(f(x)={x}^{2}\) but shifted left 5 units. The graph of \(h(x)={(x-5)}^{2}\) is the same as the graph of \(f(x)={x}^{2}\) but shifted right 5 units.
Graph \(f(x)={(x-5)}^{2}\) using a horizontal shift.

Bonisa impendulo

We first draw the graph of \(f(x)={x}^{2}\) on
the grid.
Determine h.
Shift the graph \(f(x)={x}^{2}\) to the right 5 units.
Graph \(f(x)={(x-4)}^{2}\) using a horizontal shift.

Bonisa impendulo
Graph \(f(x)={(x+6)}^{2}\) using a horizontal shift.

Bonisa impendulo
Graph \(f(x)={(x+1)}^{2}-2\) using transformations.

Bonisa impendulo

This function will involve two transformations and we need a plan.
Let’s first identify the constants h, k.

The h constant gives us a horizontal shift and the k gives us a vertical shift.

We first draw the graph of \(f(x)={x}^{2}\) on the grid.
Graph \(f(x)={(x+2)}^{2}-3\) using transformations.

Bonisa impendulo
Graph \(f(x)={(x-3)}^{2}+1\) using transformations.

Bonisa impendulo
Graph \(f(x)=3{x}^{2}.\)

Bonisa impendulo

We will graph the functions \(f(x)={x}^{2}\) and \(g(x)=3{x}^{2}\) on the same grid. We will choose a few points on \(f(x)={x}^{2}\) and then multiply the y-values by 3 to get the points for \(g(x)=3{x}^{2}.\)
Graph \(f(x)=-3{x}^{2}.\)

Bonisa impendulo
Graph \(f(x)=2{x}^{2}.\)

Bonisa impendulo

Rewrite \(f(x)=-3{x}^{2}-6x-1\) in the \(f(x)=a{(x-h)}^{2}+k\) form by completing the square.

Bonisa impendulo


Separate the x terms from the constant.
Factor the coefficient of \({x}^{2}\), \(-3\).
Prepare to complete the square.
Take half of 2 and then square it to complete the square. \({(\frac{1}{2}\cdot 2)}^{2}=1\)
The constant 1 completes the square in the parentheses, but the parentheses is multiplied by \(-3\). So we are really adding \(-3\) We must then add 3 to not change the value of the function.
Rewrite the trinomial as a square and subtract the constants.
The function is now in the \(f(x)=a{(x-h)}^{2}+k\) form.

Rewrite \(f(x)=-4{x}^{2}-8x+1\) in the \(f(x)=a{(x-h)}^{2}+k\) form by completing the square.

Bonisa impendulo

\(f(x)=-4{(x+1)}^{2}+5\)
Rewrite \(f(x)=2{x}^{2}-8x+3\) in the \(f(x)=a{(x-h)}^{2}+k\) form by completing the square.

Bonisa impendulo

\(f(x)=2{(x-2)}^{2}-5\)

Graph \(f(x)={x}^{2}+6x+5\) by using transformations.

Bonisa impendulo

Step 1. Rewrite the function in \(f(x)=a{(x-h)}^{2}+k\) vertex form by completing the square.


Separate the x terms from the constant.
Take half of 6 and then square it to complete the square. \({(\frac{1}{2}\cdot 6)}^{2}=9\)
We both add 9 and subtract 9 to not change the value of the function.
Rewrite the trinomial as a square and subtract the constants.
The function is now in the \(f(x)={(x-h)}^{2}+k\) form.

Step 2: Graph the function using transformations.

Looking at the h, k values, we see the graph will take the graph of \(f(x)={x}^{2}\) and shift it to the left 3 units and down 4 units.

We first draw the graph of \(f(x)={x}^{2}\) on the grid.

Graph \(f(x)={x}^{2}+2x-3\) by using transformations.

Bonisa impendulo
Graph \(f(x)={x}^{2}-8x+12\) by using transformations.

Bonisa impendulo

Graph \(f(x)=-2{x}^{2}-4x+2\) by using transformations.

Bonisa impendulo

Step 1. Rewrite the function in \(f(x)=a{(x-h)}^{2}+k\) vertex form by completing the square.


Separate the x terms from the constant.
We need the coefficient of \({x}^{2}\) to be one. We factor \(-2\) from the x-terms.
Take half of 2 and then square it to complete the square. \({(\frac{1}{2}\cdot 2)}^{2}=1\)
We add 1 to complete the square in the parentheses, but the parentheses is multiplied by \(-2\). Se we are really adding \(-2\). To not change the value of the function we add 2.
Rewrite the trinomial as a square and subtract the constants.
The function is now in the \(f(x)=a{(x-h)}^{2}+k\) form.

Step 2. Graph the function using transformations.

We first draw the graph of \(f(x)={x}^{2}\) on the grid.

Graph \(f(x)=-3{x}^{2}+12x-4\) by using transformations.

Bonisa impendulo
Graph \(f(x)=-2{x}^{2}+12x-9\) by using transformations.

Bonisa impendulo

ⓐ Rewrite \(f(x)=2{x}^{2}+4x+5\) in \(f(x)=a{(x-h)}^{2}+k\) form and ⓑ graph the function using properties.

Bonisa impendulo

Rewrite the function in \(f(x)=a{(x-h)}^{2}+k\) form by completing the square.	\(f(x)=2{x}^{2}+4x+5\)
	\(f(x)=2({x}^{2}+2x)+5\)
	\(f(x)=2({x}^{2}+2x+1)+5-2\)
	\(f(x)=2{(x+1)}^{2}+3\)
Identify the constants \(a,h,k.\)	\(\ a=2\ h=-1\ k=3\)
Since \(a=2\), the parabola opens upward.
The axis of symmetry is \(x=h\).	The axis of symmetry is \(x=-1\).
The vertex is \((h,k)\).	The vertex is \((-1,3)\).
Find the y-intercept by finding \(f(0)\).	\(f(0)=2⋅{0}^{2}+4⋅0+5\)
	\(f(0)=5\)
	y-intercept \((0,5)\)
Find the point symmetric to \((0,5)\) across the axis of symmetry.	\((-2,\ 5)\)
Find the x-intercepts.	The discriminant negative, so there are no x-intercepts. Graph the parabola.

ⓐ Rewrite \(f(x)=3{x}^{2}-6x+5\) in \(f(x)=a{(x-h)}^{2}+k\) form and ⓑ graph the function using properties.

Bonisa impendulo

ⓐ \(f(x)=3{(x-1)}^{2}+2\)
ⓑ
ⓐ Rewrite \(f(x)=-2{x}^{2}+8x-7\) in \(f(x)=a{(x-h)}^{2}+k\) form and ⓑ graph the function using properties.

Bonisa impendulo

ⓐ \(f(x)=-2{(x-2)}^{2}+1\)
ⓑ

Determine the quadratic function whose graph is shown.

Bonisa impendulo

Since it is quadratic, we start with the \(f(x)=a{(x-h)}^{2}+k\ \text{form.}\)
The vertex, \((h,k),\) is \((-2,-1)\) so \(h=-2\) and \(k=-1.\)	\(\ f(x)=a{(x-(-2))}^{2}-1\)
To find \(a\), we use the \(y\)-intercept, \((0,7)\).
So \(f(0)=7\).	\(\ 7=a{(0+2)}^{2}-1\)
Solve for \(a\).	\(\ 7=4a-1\)
	\(\ 8=4a\)
	\(\ 2=a\)
Write the function.	\(\ f(x)=a{(x-h)}^{2}+k\)
Substitute in \(h=-2,k=-1\) and \(a=2\).	\(\ f(x)=2{(x+2)}^{2}-1\)

Write the quadratic function in \(f(x)=a{(x-h)}^{2}+k\) form whose graph is shown.

Bonisa impendulo

\(f(x)={(x-3)}^{2}-4\)
Determine the quadratic function whose graph is shown.

Bonisa impendulo

\(f(x)={(x+3)}^{2}-1\)
\(f(x)={x}^{2},\ g(x)={x}^{2}+4,\) and \(h(x)={x}^{2}-4.\)

Bonisa impendulo

ⓐ

ⓑ The graph of \(g(x)={x}^{2}+4\) is the same as the graph of \(f(x)={x}^{2}\) but shifted up 4 units. The graph of \(h(x)={x}^{2}-4\) is the same as the graph of \(f(x)={x}^{2}\) but shift down 4 units.
\(f(x)={x}^{2},\ g(x)={x}^{2}+7,\) and \(h(x)={x}^{2}-7.\)
\(f(x)={x}^{2}+3\)

Bonisa impendulo
\(f(x)={x}^{2}-7\)

Symbols used here

\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.

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

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

|x|

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

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: Graph Quadratic Functions Using Transformations

Graph quadratic equations of the form
Graph quadratic functions of the form
Graph quadratic functions of the form
Graph quadratic functions using transformations
Find a quadratic function from its graph
If
If
If

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.

Zama ngokwakho

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

IiNkqubo Algebra

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

We first draw the graph of \(f(x)={x}^{2}\) on the grid.
Determine \(k\).

Shift the graph \(f(x)={x}^{2}\) down 3.