# 3.9: Vertex Form of a Quadratic Function

Difficulty Level: Advanced Created by: CK-12
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Practice Use Vertex Form of Quadratics

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Given the equation \begin{align*}y=3(x+4)^2+2\end{align*}, list the transformations of \begin{align*}y=x^2.\end{align*}

### Guidance

The equation for a basic parabola with a vertex at \begin{align*}(0,0)\end{align*} is \begin{align*}y=x^2\end{align*}. You can apply transformations to the graph of \begin{align*}y=x^2\end{align*} to create a new graph with a corresponding new equation. This new equation can be written in vertex form. The vertex form of a quadratic function is \begin{align*} y=a(x-h)^2+k\end{align*} where:

• \begin{align*}|a|\end{align*} is the vertical stretch factor. If \begin{align*}a\end{align*} is negative, there is a vertical reflection and the parabola will open downwards.
• \begin{align*}k\end{align*} is the vertical translation.
• \begin{align*}h\end{align*} is the horizontal translation.

Given the equation of a parabola in vertex form, you should be able to sketch its graph by performing transformations on the basic parabola. This process is shown in the examples.

#### Example A

Given the following function in vertex form, identify the transformations of \begin{align*}y=x^2\end{align*}.

\begin{align*}y=-\frac{1}{2}(x-2)^2-1\end{align*}

Solution:

• \begin{align*}a\end{align*} – Is \begin{align*}a\end{align*} negative? YES. The parabola will open downwards.
• \begin{align*}a\end{align*} – Is there a number in front of the squared portion of the equation? YES. The vertical stretch factor is the absolute value of this number. Therefore, the vertical stretch of this function is \begin{align*}\frac{1}{2}\end{align*}.
• \begin{align*}k\end{align*} – Is there a number after the squared portion of the equation? YES. The value of this number is the vertical translation. The vertical translation is –1.
• \begin{align*}h\end{align*} – Is there a number after the variable ‘\begin{align*}x\end{align*}’? YES. The value of this number is the opposite of the sign that appears in the equation. The horizontal translation is +2.

#### Example B

Given the following transformations, determine the equation of the image of \begin{align*}y=x^2\end{align*} in vertex form.

• Vertical stretch by a factor of 3
• Vertical translation up 5 units
• Horizontal translation left 4 units

Solution:

• \begin{align*}a\end{align*} – The image is not reflected in the \begin{align*}x\end{align*}-axis. A negative sign is not required.
• \begin{align*}a\end{align*} – The vertical stretch is 3, so \begin{align*}a=3\end{align*}.
• \begin{align*}k\end{align*} – The vertical translation is 5 units up, so \begin{align*}k=5\end{align*}.
• \begin{align*}h\end{align*} – The horizontal translation is 4 units left so \begin{align*}h=-4\end{align*}.

The equation of the image of \begin{align*}y=x^2\end{align*} is \begin{align*}y=3(x+4)^2+5\end{align*}.

#### Example C

Using \begin{align*}y=x^2\end{align*} as the base function, identify the transformations that have occurred to produce the following image graph. Use these transformations to write the equation in vertex form.

Solution:

\begin{align*}a\end{align*} – The parabola does not open downward so \begin{align*}a \end{align*} will be positive.

\begin{align*}a\end{align*} – The \begin{align*}y\end{align*}-values of 1 and 4 are now up 3 and up 12. \begin{align*}a = 3\end{align*}.

\begin{align*}k\end{align*} – The \begin{align*}y-\end{align*}coordinate of the vertex is –5 so \begin{align*}k=-5\end{align*}.

\begin{align*}h\end{align*} – The \begin{align*}x-\end{align*}coordinate of the vertex is +3 so \begin{align*}h=3\end{align*}.

The equation is \begin{align*}y=3(x-3)^2-5\end{align*}.

#### Example D

In general, the mapping rule used to generate the image of a function is \begin{align*}(x,y) \rightarrow (x^\prime,y^\prime)\end{align*} where \begin{align*}(x^\prime,y^\prime)\end{align*} are the coordinates of the image graph. The resulting mapping rule from \begin{align*}y=x^2\end{align*} to the image \begin{align*}y=a(x-h)^2+k\end{align*} is \begin{align*}(x,y) \rightarrow (x+h,ay+k)\end{align*}. The mapping rule details the transformations that were applied to the coordinates of the base function \begin{align*}y=x^2\end{align*}.

Given the following quadratic equation, \begin{align*}y=2(x+3)^2+5\end{align*} write the mapping rule and create a table of values for the mapping rule.

Solution:

The mapping rule for this function will tell exactly what changes were applied to the coordinates of the base quadratic function.

\begin{align*}y=2(x+3)^2+5: \quad (x,y) \rightarrow (x-3,2y+5)\end{align*}

These new coordinates of the image graph can be plotted to generate the graph.

#### Concept Problem Revisited

Given the equation \begin{align*}y=3(x+4)^2+2\end{align*}, list the transformations of \begin{align*}y=x^2\end{align*}.

\begin{align*}a=3\end{align*} so the vertical stretch is 3. \begin{align*}k=2\end{align*} so the vertical translation is up 2. \begin{align*}h=-4\end{align*} so the horizontal translation is left 4.

### Vocabulary

Horizontal translation
The horizontal translation is the change in the base graph \begin{align*}y=x^2\end{align*} that shifts the graph right or left. It changes the \begin{align*}x-\end{align*}coordinate of the vertex.
Mapping Rule
The mapping rule defines the transformations that have occurred to a function. The mapping rule is \begin{align*}(x,y) \rightarrow (x^\prime,y^\prime)\end{align*} where \begin{align*}(x^\prime,y^\prime)\end{align*} are the coordinates of the image graph.
Transformation
A transformation is any change in the base graph \begin{align*}y=x^2\end{align*}. The transformations that apply to the parabola are a horizontal translation, a vertical translation, a vertical stretch and a vertical reflection.
Vertex form of \begin{align*}y = x^2\end{align*}
The vertex form of \begin{align*}y = x^2\end{align*} is the form of the quadratic base function \begin{align*}y=x^2\end{align*} that shows the transformations of the image graph. The vertex form of the equation is \begin{align*} y=a(x-h)^2+k\end{align*}.
Vertical Reflection
The vertical reflection is the reflection of the image graph in the \begin{align*}x\end{align*}-axis. The graph opens downward and the \begin{align*}y\end{align*}-values are negative values.
Vertical Stretch
The vertical stretch is the change made to the base function \begin{align*}y=x^2\end{align*} by stretching (or compressing) the graph vertically. The vertical stretch will produce an image graph that appears narrower (or wider) then the original base graph of \begin{align*}y=x^2\end{align*}.
Vertical Translation
The vertical translation is the change in the base graph \begin{align*}y=x^2\end{align*} that shifts the graph up or down. It changes the \begin{align*}y-\end{align*}coordinate of the vertex.

### Guided Practice

1. Identify the transformations of \begin{align*}y=x^2\end{align*} for the quadratic function \begin{align*}-2(y+3)=(x-4)^2\end{align*}

2. List the transformations of \begin{align*}y=x^2\end{align*} and graph the function \begin{align*}=-(x+5)^2+4\end{align*}

3. Graph the function \begin{align*}y=2(x-2)^2+3\end{align*} using the mapping rule method.

1. Rewrite the equation in vertex form. \begin{align*}a\end{align*}\begin{align*}a\end{align*} is negative so the parabola opens downwards.

\begin{align*}a\end{align*} – The vertical stretch of this function is \begin{align*}\frac{1}{2}\end{align*}.

\begin{align*}k\end{align*} – The vertical translation is -3.

\begin{align*}h\end{align*} – The horizontal translation is +4.

2.

\begin{align*}a & \rightarrow negative\\ a & \rightarrow 1\\ k & \rightarrow +4\\ h & \rightarrow -5\end{align*}

3. Mapping Rule \begin{align*}(x,y) \rightarrow (x+2,2y+3)\end{align*}

Make a table of values:

\begin{align*}x \rightarrow\end{align*} \begin{align*}x+2\end{align*} \begin{align*}y \rightarrow\end{align*} \begin{align*}2y+3\end{align*}
–3 –1 9 21
–2 0 4 11
–1 1 1 5
0 2 0 3
1 3 1 5
2 4 4 11
3 5 9 21

Draw the graph:

### Practice

Identify the transformations of \begin{align*}y=x^2\end{align*} in each of the given functions:

1. \begin{align*}y=4(x-2)^2-9\end{align*}
2. \begin{align*}y=-\frac{1}{6}x^2+7\end{align*}
3. \begin{align*}y=-3(x-1)^2-6\end{align*}
4. \begin{align*}y=\frac{1}{5}(x+4)^2+3\end{align*}
5. \begin{align*}y=5(x+2)^2\end{align*}

1. \begin{align*}y=2(x-4)^2-5\end{align*}
2. \begin{align*}y=-\frac{1}{3}(x-2)^2+6\end{align*}
3. \begin{align*}y=-2(x+3)^2+7\end{align*}
4. \begin{align*}y=-\frac{1}{2}(x+6)^2+9\end{align*}
5. \begin{align*}y=\frac{1}{3}(x-4)^2\end{align*}

Using the following mapping rules, write the equation, in vertex form, that represents the image of \begin{align*}y = x^2\end{align*}.

1. \begin{align*}(x,y) \rightarrow \left(x+1, -\frac{1}{2}y\right)\end{align*}
2. \begin{align*}(x,y) \rightarrow (x+6,2y-3)\end{align*}
3. \begin{align*}(x,y) \rightarrow \left(x-1, \frac{2}{3}y+2\right)\end{align*}
4. \begin{align*}(x,y) \rightarrow (x+3,3y+1)\end{align*}
5. \begin{align*}(x,y) \rightarrow \left(x-5,-\frac{1}{3}y-7\right)\end{align*}

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Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
Absolute Value The absolute value of a number is the distance the number is from zero. Absolute values are never negative.
quadratic function A quadratic function is a function that can be written in the form $f(x)=ax^2 + bx + c$, where $a$, $b$, and $c$ are real constants and $a\ne 0$.
Vertex form The vertex form of a parabola is $(x-h)^2=4p(y-k)$ or $(y-k)^2=4p(x-h)$ where $(h, k)$ is the vertex.

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