Graph Quadratic Functions Using Transformations

Graph Quadratic Functions of the Form $f(x)=x^2+k$

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 9.53. 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.

Now that we have seen the effect of the constant, $k$, it is easy to graph functions of the form $f(x)=x^2+k$. We just start with the basic parabola of $f(x)=x^2$ and then shift it up or down.

It may be helpful to practice sketching $f(x)=x^2$ quickly. We know the values and can sketch the graph from there.

Once we know this parabola, it will be easy to apply the transformations. The next example will require a vertical shift.

Example 9.54. Graph $f(x)=x^2-3$ using a vertical shift.

Solution.

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 Quadratic Functions of the Form $f(x)=(x-h)^2$

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 9.55. 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 graph of $g(x)=(x-1)^2$ is the same as the graph of $f(x)=x^2$ but shifted right 1 unit.

The graph of $h(x)=(x+1)^2$ is the same as the graph of $f(x)=x^2$ but shifted left 1 unit.

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 9.56. 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.

Now that we know the effect of the constants $h$ and $k$, we will graph a quadratic function of the form $f(x)=(x-h)^2+k$ by first drawing the basic parabola and then making a horizontal shift followed by a vertical shift. We could do the vertical shift followed by the horizontal shift, but most students prefer the horizontal shift followed by the vertical.

Example 9.57. Graph $f(x)=(x+1)^2-2$ using transformations.

Solution.

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. To graph $f(x)=(x+1)^2$, shift the graph $f(x)=x^2$ to the left 1 unit. To graph $f(x)=(x+1)^2-2$, shift the graph $f(x)=(x+1)^2$ down 2 units.

Graph Quadratic Functions of the Form $f(x)=ax^2$

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 $f(x)=ax^2$.

Let’s look at the quadratic functions $f(x)=x^2$, $g(x)=2x^2$, and $h(x)=\tfrac12x^2$.

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

The graph of the function $g(x)=2x^2$ is “skinnier” than the graph of $f(x)=x^2$.

The graph of the function $h(x)=\tfrac12x^2$ is “wider” than the graph of $f(x)=x^2$.

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 9.58. Graph $f(x)=3x^2$.

Solution.

We will graph the functions $f(x)=x^2$ and $g(x)=3x^2$ 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 $g(x)=3x^2$.

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 $f(x)=ax^2$ affect their graphs. We can now put this together and graph quadratic functions $f(x)=ax^2+bx+c$ 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 $x^2$ 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 9.59. Rewrite $f(x)=-3x^2-6x-1$ in the $f(x)=a(x-h)^2+k$ form by completing the square.

Solution.

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.

Example 9.60. Graph $f(x)=x^2+6x+5$ by using transformations.

Solution.

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

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. To graph $f(x)=(x+3)^2$, shift the graph $f(x)=x^2$ to the left 3 units. To graph $f(x)=(x+3)^2-4$, shift the graph $f(x)=(x+3)^2$ down 4 units.

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

Example 9.61. Graph $f(x)=-2x^2-4x+2$ by using transformations.

Solution.

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

Step 2. Graph the function using transformations.

We first draw the graph of $f(x)=x^2$ on the grid. To graph $f(x)=-2x^2$, multiply the $y$-values in the parabola of $f(x)=x^2$ by $-2$. To graph $f(x)=-2(x+1)^2$, shift the graph $f(x)=-2x^2$ to the left 1 unit. To graph $f(x)=-2(x+1)^2+4$, shift the graph $f(x)=-2(x+1)^2$ up 4 units.

Now that we have completed the square to put a quadratic function into $f(x)=a(x-h)^2+k$ form, we can also use this technique to graph the function using its properties as in the previous section.

If we look back at the last few examples, we see that the vertex is related to the constants $h$ and $k$.

In each case, the vertex is $(h,k)$. Also the axis of symmetry is the line $x=h$.

We rewrite our steps for graphing a quadratic function using properties for when the function is in $f(x)=a(x-h)^2+k$ form.

Example 9.62. (a) Rewrite $f(x)=2x^2+4x+5$ in $f(x)=a(x-h)^2+k$ form and (b) graph the function using properties.

Solution.

Rewrite the function in $f(x)=a(x-h)^2+k$ form by completing the square.

Identify the constants $a=2$, $h=-1$, $k=3$. Since $a=2$, the parabola opens upward. The axis of symmetry is $x=-1$. The vertex is $(-1,3)$.

Find the $y$-intercept by finding $f(0)$.

so the $y$-intercept is $(0,5)$. The point symmetric to $(0,5)$ across the axis of symmetry is $(-2,5)$. The discriminant is negative, so there are no $x$-intercepts. Graph the parabola.

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 9.63. Determine the quadratic function whose graph is shown.

Solution.

Since it is quadratic, we start with the $f(x)=a(x-h)^2+k$ form.

The vertex, $(h,k)$, is $(-2,-1)$ so $h=-2$ and $k=-1$. To find $a$, we use the $y$-intercept, $(0,7)$. So $f(0)=7$.