The Coordinates of the Vertex

7.15.3 The Coordinates of the Vertex

7.15.3 • The Coordinates of the Vertex

Activity

For questions 1 – 2, use the Desmos graphing tool or technology outside the course.

1. Graph $y = x^{2}$ . Then, add different numbers to $x$ before it is squared (for example, $y = (x + 4)^{2}$ , $y = (x - 3)^{2}$ ) and observe how the graph changes.

Compare your answer:

Adding a number to $x$ moves the graph to the left. Subtracting a number (or adding a negative number) moves the graph to the right. The vertex stays on the $x$ -axis.

2. Graph $y = (x - 1)^{2}$ . Then, experiment with each of the following changes to the function and see how they affect the graph and the vertex.

a. Add different constant terms to $(x - 1)^{2}$ (for example: $y = (x - 1)^{2} + 5$ , $y = (x - 1)^{2} - 9)$ .

Compare your answer:

Adding a constant term moves it up and subtracting moves it down. The $x$ -coordinate of the vertex doesn't change.

b. Multiplying $(x - 1)^{2}$ by different coefficients (for example: $y = 3 (x - 1)^{2}$ , $y = - 2 (x - 1)^{2}$ , $y = 0.5 (x - 1)^{2}$ .

Compare your answer:

Multiplying the squared term by a number doesn't change the vertex, but it changes the opening of the graph to "wider" or "narrower," or changes the direction of the graph to open upward or downward.

3. Without graphing, find the coordinates of the vertex of $y = (x + 10)^{2}$ and determine if it opens up or down.

a. What is the $x$ -coordinate of the vertex?

The $x$ -coordinate of the vertex of this equation is -10.

b. What is the $y$ -coordinate of the vertex?

The $y$ -coordinate of the vertex of this equation is 0.

c. Does the graph open up or down?

The graph opens upward. The squared term has a positive coefficient (1).

Select the solution button to see the solutions to question 3 organized into a table.

Compare your answer:

Equation	Coordinates of vertex	Graph opens up or down?
$y = (x + 10)^{2}$	$(- 10, 0)$	Up

4. Without graphing, find the coordinates of the vertex of $y = (x - 4)^{2} + 8$ and determine if it opens up or down.

a. What is the $x$ -coordinate of the vertex?

The $x$ -coordinate of the vertex of this equation is 4.

b. What is the $y$ -coordinate of the vertex?

The $y$ -coordinate of the vertex of this equation is 8.

c. Does the graph open up or down?

The graph opens upward. The squared term has a positive coefficient (1).

Select the solution button to see the solutions to questions 3 – 4 organized into a table.

Compare your answer:

Equation	Coordinates of vertex	Graph opens up or down?
$y = (x + 10)^{2}$	$(- 10, 0)$	Up
$y = (x - 4)^{2} + 8$	$(4, 8)$	Up

What patterns do you notice? Can you use those ideas to find the vertices for the next equation? What do you know about whether it will open up or down?

5. Without graphing, find the coordinates of the vertex of $y = - (x - 4)^{2} + 8$ and determine if it opens up or down.

a. What is the $x$ -coordinate of the vertex?

The $x$ -coordinate of the vertex of this equation is 4.

b. What is the $y$ -coordinate of the vertex?

The $y$ -coordinate of the vertex of this equation is 8.

c. Does the graph open up or down?

The graph opens downward. The squared term has a negative coefficient (-1).

Select the solution button to see the solutions to questions 3 – 5 organized into a table

Compare your answer:

Equation	Coordinates of vertex	Graph opens up or down?
$y = (x + 10)^{2}$	$(- 10, 0)$	Up
$y = (x - 4)^{2} + 8$	$(4, 8)$	Up
$y = - (x - 4)^{2} + 8$	$(4, 8)$	Down

6. The next equation, $y = x^{2} - 7$ , looks a little different from the previous ones. Can you use what you’ve learned to find the coordinates of its vertex and determine if it opens up or down.

a. What is the $x$ -coordinate of the vertex?

The $x$ -coordinate of the vertex of this equation is 0.

b. What is the $y$ -coordinate of the vertex?

The $y$ -coordinate of the vertex of this equation is -7.

c. Does the graph open up or down?

The graph opens upward. The squared term has a positive coefficient $(1)$ .

Select the solution button to see the solutions to questions 3 – 6 organized into a table

Compare your answer:

Equation	Coordinates of vertex	Graph opens up or down?
$y = (x + 10)^{2}$	$(10, 0)$	Up
$y = (x - 4)^{2} + 8$	$(4, 8)$	Up
$y = - (x - 4)^{2} + 8$	$(4, 8)$	Down
$y = x^{2} - 7$	$(0, - 7)$	Up

7. Are the patterns you noticed still happening? See if you can apply those patterns to the following equations.

a. For the equation $y = \frac{1}{2} (x + 3)^{2} - 5$ , what is the $x$ -coordinate of the vertex?

The $x$ -coordinate of the vertex of this equation is $- 3$ .

b. For the equation $y = \frac{1}{2} (x + 3)^{2} - 5$ , what is the $y$ -coordinate of the vertex?

The $y$ -coordinate of the vertex of this equation is $- 5$ .

c. For the equation $y = \frac{1}{2} (x + 3)^{2} - 5$ , does the graph open up or down?

The graph opens upward. The squared term has a positive coefficient $\frac{1}{2})$ .

d. For the equation $y = - (x + 100)^{2} + 50$ , what is the $x$ -coordinate of the vertex?

The $x$ -coordinate of the vertex of this equation is $- 100$ .

e. For the equation $y = - (x + 100)^{2} + 50$ , what is the $y$ -coordinate of the vertex?

The $y$ -coordinate of the vertex of this equation is $50$ .

f. For the equation $y = - (x + 100)^{2} + 50$ , does the graph open up or down?

The graph opens downward. The squared term has a negative coefficient $(- 1)$ .

Select the solution button to see the solutions to questions 3 – 7 organized into a table.

Compare your answer:

Equation	Coordinates of vertex	Graph opens up or down?
$y = (x + 10)^{2}$	$(- 10, 0)$	Up
$y = (x - 4)^{2} + 8$	$(4, 8)$	Up
$y = - (x - 4)^{2} + 8$	$(4, 8)$	Down
$y = x^{2} - 7$	$(0, - 7)$	Up
$y = 12 (x + 3)^{2} - 5$	$(- 3, - 5)$	Up
$y = - (x + 100)^{2} + 50$	$(- 100, 50)$	Down
$y = a (x + m)^{2} + n$	a. _____ , b. _____	c. _____

The table now has all the equations from questions 3 – 7, plus an additional row. Use this information to answer question 8. Do the patterns you noticed help you fill in the last row?

8. Use the completed rows of the table to hypothesize about the coordinates of the vertex of the equation in the last row, and whether the parabola opens up or down.

a. For the equation $y = a (x + m)^{2} + n$ , what is the $x$ -coordinate of the vertex?

The $x$ -coordinate of the vertex of this equation is – $m$ .

b. For the equation $y = a (x + m)^{2} + n$ , what is the $y$ -coordinate of the vertex?

The $y$ -coordinate of the vertex of this equation is $n$ .

c. For the equation $y = a (x + m)^{2} + n$ , can you determine if the graph opens up or down with the information you are given?

No, because the direction of opening depends on whether $a$ is positive or negative.

Are you ready for more?

Extending Your Thinking

Examine the graph below, then use it to answer questions 1 – 3.

Graph of a parabola on a coordinate plane. The x-axis has a scale of 1 extending from negative 5 to 8. The vertical-axis extends from negative 6 to 18 with a scale of 2. Three points on the parabola are labeled: (0, 14), (2, negative 2), and (5, 4).

What is the $x$ -coordinate of the vertex of this graph?

Compare your answer: 3

What is the $y$ -coordinate of the vertex of this graph?

Compare your answer: -4

Which quadratic equation best fits the function in the graph?

Compare your answer:

Answer: $y = 2 (x - 3)^{2} - 4$ . The equation $y = (x - 3)^{2} - 4$ has the vertex at $(3, - 4)$ . The graph shown has a $y$ -intercept at (0,14) which doesn't work for the equation $y = (x - 3)^{2} - 4$ . The equation $y = 2 (x - 3)^{2} - 4$ appears to fit. When $x = 0$ the $y$ -value is $2 (0 - 3)^{2} - 4 = 14$ .

Video: Learning About the Vertex Form of a Quadratic

Watch the following video to learn more about the vertex form of a quadratic.

Access multimedia content

Self Check

Find the vertex for

y=3(x-2)^2-5

$(2, 5)$
$(-2, -5)$
$(2, -5)$
$(3, -5)$

Additional Resources

Using Vertex Form

The vertex form of a quadratic is below:

Equation of a parabola in vertex form is labeled with descriptions and color coded. y equals a times the quantity of x minus h squared plus k. A line from the parameter "a" leads to a textbox that describes that it impacts the parabola's vertical reflection if the a is negative, vertical stretch if the a is greater than 1, or vertical compression if the value of a is between 0 and 1. A line from the parameter "h" leads to a textbox that describes how h indicates either a horizontal shift to the right, if h is added, or to the left, if h is subtracted. The final parameter k has a line that leads to a textbox describing how it determines the vertical shift of the function. If the value of k is added, then the parabola is shifted up. If the value of k is subtracted, then the parabola is shifted down.

Notice that the value of $h$ is the opposite of what it is in the parenthesis. This seems odd: when we add $k$ , the vertex of the parabola moves up $k$ units. But when we subtract $h$ within the squared term, the vertex of the parabola moves right $h$ units, not left.

Let’s explore why. We know the $y$ -coordinate of the vertex is $y = k$ . To find that using this equation, we need to know where $a (x - h)^{2}$ is zero, because that will make $y = k$ . Set $a (x - h)^{2} = 0$ . This happens when $x - h = 0$ , which is when $x = h$ . That means $h$ is the $x$ -coordinate of the vertex.

Example

Find the vertex of $y = 3 (x - 4)^{2} - 2$ .

The parabola of the quadratic will shift right 4 (remember this horizontal shift is counterintuitive and translates in the opposite direction of what the expression “looks like”) and down 2.

So, the vertex is at $(4, - 2)$ .

Try it

Using Vertex Form

Find the vertex of $y = (x - 2)^{2} + 5$ .

Here is how to find the vertex:

The quadratic shifts to the right 2 and up 5. The vertex is at $(2, 5)$ .