**9.7 Graph Quadratic Functions Using Transformations**

The topics covered in this section are:

- Graph quadratic equations of the form $f(x)=x^{2}+k$
- Graph quadratic functions of the form $f(x)=(x-h)^{2}$
- Graph quadratic functions of the form $f(x)=ax^{2}$
- Graph quadratic functions using transformations
- Find a quadratic function from its graph

**9.7.1 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 1**

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.*

**GRAPH A QUADRATIC FUNCTION OF THE FORM $f(x)=x^{2}$ 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.

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 2**

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 $k$. | $f(x)=x^{2}+k$ $f(x)=x^{2}-3$ |

$k=-3$ | |

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

**9.7.2 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 3**

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*.

**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 right $h$ units.
- If $h<0$, shift the parabola horizontally left $|h|$ units.

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 4**

Graph $f(x)=(x-6)^{2}$ using a horizontal shift.

**Solution**

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

Determine $h$. | $f(x)=(x-h)^{2}$ $f(x)=(x-6)^{2}$ |

$h=6$ | |

Shift the graph $f(x)=x^{2}$ to the right $6$ 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 5**

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 $h,k$.

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

$f(x)=(x-h)^{2}+k$

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

$h=-1$ and $k=-2$

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.

**9.7.3 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 $f(x)=ax^{2}$.

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 $a$.

**GRAPH OF A QUADRATIC FUNCTION OF THE FORM $f(x)=ax^{2}$**

The coefficient $a$ in the function $f(x)=ax^{2}$ affects the graph of $f(x)=x^{2}$ by stretching or compressing it.

- If $0<|a|<1$, the graph of $f(x)=ax^{2}$ will be “wider” than the graph of $f(x)=x^{2}$.
- If $|a| >1$, the graph of $f(x)=ax^{2}$ will be “skinnier” than the graph of $f(x)=x^{2}$.

**Example 6**

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 the multiply the $y$-values by $3$ to get the points for $g(x)=3x^{2}$.

**9.7.4 Graph Quadratic Functions Using Transformations**

We have learned how the constants $a,h,$ and $k$ in the functions, $f(x)=x^{2}, f(x)=(x-h)^{2},$ 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 7**

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

**Solution**

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

Separate the $x$ terms from the constant. | $f(x)=-3x^{+2}-6x \ \ \ \ -1$ |

Factor the coefficient of $x^{2}, -3$ | $f(x)=-3(x^{2}+2x) \ \ \ \ -1$ |

Prepare to complete the square. | $f(x)=-3(x^{2}+2x \ \ \ \ )-1$ |

Take half of $2$ and the 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. | $f(x)=-3(x+1)+2$ |

The function is now in the $f(x)=a(x-h)^{2}+k$ form. | $f(x)=a(x-h)^{2} +k$ $f(x)=-3(x+1)^{2}+2$ |

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 8**

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

**Solution**

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

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

Seperate the $x$ terms from the constant. | $f(x)=x^{2}+6x+5$ |

Take half of $6$ and the 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$ | $f(x)=x^{2}+6x+9+5-9$ |

Rewrite the trinomial as a square and subtract the constants. | $f(x)=(x+3)^{2}-4$ |

The function is now in the $f(x)=(x-h)^{2}+k$ form. | $f(x)=(x-h)^{2}+k$ $f(x)=(x+3)^{2}-4$ |

**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.

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

**HOW TO: Graph a quadratic function using transformations.**

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

**Example 9**

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.

$f(x)=-2x^{2}-4x+2$ | |

Separate the $x$ terms from the constant. | $f(x)=-2x^{2}-4x+2$ |

We need the coefficient of $x^{2}$ to be one. We factor $-2$ from the $x$-terms. | $f(x)=-2(x^{2}+2x)+2$ |

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, bu the parentheses is multiplied by $-2$. So we are really adding $-2$. To not change the value of the function we add $2$. | $f(x)=-2(x^{2}+2x+1)+2+2$ |

Rewrite the trinomial as a square and subtract the constants. | $f(x)=-2(x+1)^{2}+4$ |

The function is now in the $f(x)=a(x-h)^{2}+k$ form. | $f(x)=a(x-h)^{2}+k$ $f(x)=-2(x+1)^{2}+4$ |

**Step 2.** Graph the function using transformations.

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

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.

**HOW TO: Graph a quadratic function in the form $f(x)=a(x-h)^{2}+k$ using properties.**

- Rewrite the function in $f(x)=a(x-h)^{2}+k$ form.
- Determine whether the parabola opens upward, $a>0$, or downward, $a<0$.
- Find the axis of symmetry, $x=h$
- Find the vertex, $(h,k)$.
- Find the $y$-intercept. Find the point symmetric to the
*y*-intercept across the axis of symmetry. - Find the $x$-intercepts.
- Graph the parabola.

**Example 10**

- Rewrite $f(x)=2x^{2}+4x+5$ in $f(x)=a(x-h)^{2}+5$ form
- Graph the function using properties.

**Solution**

Rewrite the function $f(x)=a(x-h)^{2}+k$ form by completing the square. | $f(x)=2x^{2}+4x+5$ |

$f(x)=2(x^{2}+2x)+5$ | |

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

$f(x)=2(x+1)^{2}+3$ | |

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 $f(0)$ | $f(0)=2 \cdot 0^{2}+4 \cdot 0 +5$ |

$f(0)=5$ | |

$y$-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. |

**9.7.5 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 11**

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 $k=-1$. | $f(x)=a(x-(-2))^{2}-1$ |

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 $a=2$ | $f(x)=2(x+2)^{2}-1$ |

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*Marecek, L., & Mathis, A. H. (2020). Graph Quadratic Functions Using Transformations. In Intermediate Algebra 2e. OpenStax. https://openstax.org/books/intermediate-algebra-2e/pages/9-7-graph-quadratic-functions-using-transformations*.*License: CC BY 4.0. Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction*