How to Graph a Parabola in Vertex Form

How to Graph a Parabola in Vertex Form

Step 1: Use the vertex form provided to determine the vertex of the function. Graph this point. The general vertex form of a function is {eq}f(x)=a(x-h)^2+k {/eq}. The values that are in the place of {eq}h {/eq} and {eq}k {/eq} give us the vertex of the function. The vertex will be at the point {eq}(h,k) {/eq}.

Step 2: Create a table with two {eq}x {/eq} values that are less than the {eq}x {/eq} value of the vertex and two {eq}x {/eq} values that are greater than the {eq}x {/eq} value of the vertex. This would give a total of five values of {eq}x {/eq} including the vertex.

Step 3: Solve for the corresponding values of {eq}f(x) {/eq} using the chosen values of {eq}x {/eq} to finish the table.

Step 4: Graph the points from the table and connect these points with a smooth curve.

How to Graph a Parabola in Vertex Form Vocabulary

Vertex Form of a Quadratic Function: The equation {eq}f(x)=a(x-h)^2+k {/eq} is the vertex form of a quadratic function. When a quadratic function is given in this form we can immediately see the coordinates of the vertex of the function. The vertex will be given by {eq}(h,k) {/eq}.

Vertex: This is either the highest point or the lowest point of a parabola. If the parabola opens up the vertex will be the lowest point, if the parabola opens down the vertex will be the highest point. The vertex is the center of the parabola, the left of the vertex will be a mirror image of the right of the vertex.

So, let's try using these steps to graph a parabola in vertex form, in the following two examples.

How to Graph a Parabola in Vertex Form: Example 1

Graph the function {eq}f(x)=-3(x+2)^2+4 {/eq}

Step 1: Use the vertex form provided to determine the vertex of the function. Graph this point.

The general vertex form of a function is {eq}f(x)=a(x-h)^2+k {/eq}. The values that are in the place of {eq}h {/eq} and {eq}k {/eq} give us the vertex of the function, the vertex will be {eq}(h,k) {/eq}.

In this case {eq}h=-2 {/eq} and {eq}k=4 {/eq}, that gives us the vertex of the function as {eq}(-2,4) {/eq}. This point is graphed on the coordinate plane below.

Step 2: Create a table with two {eq}x {/eq} values that are less than the {eq}x {/eq} value of the vertex and two {eq}x {/eq} values that are greater than the {eq}x {/eq} value of the vertex. This would give a total of five values of {eq}x {/eq} including the vertex.

Step 3: Solve for the corresponding values of {eq}f(x) {/eq} using the chosen values of {eq}x {/eq} to finish the table. Use the first value of {eq}x {/eq}in the table. Substitute this value into the function to determine the output of the {eq}x {/eq} value. Simplify the resulting expression.

$$\begin{align} f(-4)&=-3(-4+2)^2+4\\ f(-4)&=-3(-2)^2+4\\ f(-4)&=-3(4)+4\\ f(-4)&=-12+4\\ f(-4)&=-8 \end{align} $$

Repeat this process with the rest of the values of {eq}x {/eq} from the table.

$$\begin{align} f(-3)&=-3(-3+2)^2+4\\ f(-3)&=-3(-1)^2+4\\ f(-3)&=-3(1)+4\\ f(-3)&=-3+4\\ f(-3)&=1 \end{align} $$

$$\begin{align} f(-1)&=-3(-1+2)^2+4\\ f(-1)&=-3(1)^2+4\\ f(-1)&=-3(1)+4\\ f(-1)&=-3+4\\ f(-1)&=1 \end{align} $$

$$\begin{align} f(0)&=-3(0+2)^2+4\\ f(0)&=-3(2)^2+4\\ f(0)&=-3(4)+4\\ f(0)&=-12+4\\ f(0)&=-8 \end{align} $$

We could now fill up the table with the corresponding values of {eq}y {/eq}.

Step 4: Graph the points from the table and connect these points with a smooth curve.

The figures below show the scatterplot and corresponding graph.

How to Graph a Parabola in Vertex Form: Example 2

Graph the function {eq}f(x)=2(x-5)^2-1 {/eq}

Step 1: Use the vertex form provided to determine the vertex of the function. Graph this point.

For the given function {eq}h=5 {/eq} and {eq}k=-1 {/eq}, that gives us the vertex of the function as {eq}(5,-1) {/eq}. This point is graphed on the coordinate plane below.

Step 2: Create a table with two {eq}x {/eq} values that are less than the {eq}x {/eq} value of the vertex and two {eq}x {/eq} values that are greater than the {eq}x {/eq} value of the vertex. This would give a total of five values of {eq}x {/eq} including the vertex.

Step 3: Solve for the corresponding values of {eq}f(x) {/eq} or {eq}y {/eq} using the values of {eq}x {/eq} in the table.

$$\begin{align} f(3)&=2(3-5)^2-1\\ f(3)&=2(-2)^2-1\\ f(3)&=2(4)-1\\ f(3)&=8-1\\ f(3)&=7 \end{align} $$

$$\begin{align} f(4)&=2(4-5)^2-1\\ f(4)&=2(-1)^2-1\\ f(4)&=2(1)-1\\ f(4)&=2-1\\ f(4)&=1 \end{align} $$

$$\begin{align} f(6)&=2(6-5)^2-1\\ f(6)&=2(1)^2-1\\ f(6)&=2(1)-1\\ f(6)&=2-1\\ f(6)&=1 \end{align} $$

$$\begin{align} f(7)&=2(7-5)^2-1\\ f(7)&=2(2)^2-1\\ f(7)&=2(4)-1\\ f(7)&=8-1\\ f(7)&=7 \end{align} $$

Filling up the table, we would have

Step 4: Graph the points from the table and connect these points with a smooth curve.