Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

While answering the questions below, plot each point on a coordinate plane with grid paper by hand. Do not use technology.

For questions 1 – 4, use the the function $f (x) = (x - 2) (x + 4) .$

1.

Find the $x$ -coordinates of the $x$ -intercepts of $f (x)$ .

Compare your answer:

$x = - 4$ and $x = 2$

2.

Find the coordinate of the vertex.

Compare your answer:

$(- 1, - 5)$

3.

On your graph, sketch a quadratic through the key features on your graph and draw a vertical line through the vertex using a different color.

Compare your answer:

A graph of a parabola opening upward with x-intercepts at (negative 4, 0) and (2, 0) and a vertex at (negative 1, negative 9), all points marked with labels.

4.

What is the equation of the vertical line that goes through the vertex?

Compare your answer:

$x = - 1$

5.

What does the vertical line that goes through the vertex do to the parabola?

Compare your answer:

Your answer may vary, but here is a sample. The vertical line $x = - 1$ cuts the parabola in half and creates two symmetrical parts.

This imaginary vertical line that goes through the vertex and cuts the parabola in half is called the axis of symmetry.

6.

On your graph, label your $x$ -intercepts, vertex, and axis of symmetry. Identify your vertex as a maximum or minimum.

Compare your answer:

Graph of a parabola opening upwards, labeled with its x-intercepts, axis of symmetry (vertical green dashed line at x equals negative 1), and vertex (minimum point) at the bottom center.

7.

Given a vertex of $(2, - 4)$ and a point $(1, 3)$ , what is another point on the quadratic?

(Hint: It is helpful to draw a picture and use the axis of symmetry).

Compare your answer:

Your answer may vary, but here is a sample.

A Cartesian plane with three labeled points: (negative 5, 3), (negative 2, 4), and (1, 3). A green dashed vertical line passes through x equals negative 2.

Since $(- 2, 4)$ is the vertex, then $x = - 2$ is the axis of symmetry. If $(1, 3)$ is 3 units away from the axis of symmetry, there is another point 3 units to the left of the axis of symmetry. This is the point $(- 5, 3)$ .

Self Check

What is the equation of the axis of symmetry for

f(x)=(x+1)(x-3)

?

$x=-4$
$x=3$
$x=-1$
$x=1$

Additional Resources

Axis of Symmetry of a Quadratic

In the graph of a quadratic function, the axis of symmetry is an imaginary line that cuts the parabola in half to create two symmetrical sides. It goes through the vertex of the parabola so the equation of the axis of symmetry is the same as the $x$ -coordinate of the vertex.

Two graphs: Figure 1 shows an upward parabola for f of x equal to x squared plus 4 times x plus 3. There is a dashed blue line bisecting the parabola through the vertex. Figure 2 shows a downward parabola for f of x equal to negative x squared plus 4 times x plus 3. There is a dashed blue line bisecting the parabola through the vertex.

Notice that in Figure 1, the vertex is at $(- 2, - 2)$ and the axis of symmetry is at $x = - 2$ .

In Figure 2, the vertex is at $(2, 7)$ and the axis of symmetry is at $x = 2$ .

Example

Find the vertex and axis of symmetry for $f (x) = (x + 2) (x - 6)$ .

Step 1 - Find the $x$ -intercepts.

Set each factor equal to 0.

$x + 2 = 0$

$x = - 2$

$x - 6 = 0$

$x = 6$

Step 2 - Find the $x$ -coordinate of the vertex.

The vertex is halfway between the two $x$ -intercepts.

$x = 6 + (- 2)^{2} = 2$

$x = \frac{6 + (- 2)}{2} = 2$

Step 3 - Find the $y$ -coordinate of the vertex.

Substitute the $x$ -coordinate into $f (x)$ .

$f (2) = (2 + 2) (2 - 6) = (4) (- 4) = - 6$

The vertex is at $(2, - 16)$ .

Step 4 - Find the axis of symmetry.

The axis of symmetry is the equation where $x$ equals the $x$ -coordinate of the function.

$x = 2$

Try it

Try It: Axis of Symmetry of a Quadratic

For questions 1 - 2, use the equation of the quadratic, $f (x) = (x - 4) (x - 10)$ .

1.

What is the vertex of the quadratic?

Compare your answer:

$(7, - 9)$

Step 1 - Find the $x$ -intercepts. Set each factor equal to 0. $x - 4 = 0$ $x = 4$ $x - 10 = 0$ $x = 10$

Step 2 - Find the $x$ -coordinate of the vertex. The vertex is halfway between the two $x$ -intercepts. $x = \frac{4 + 10}{2} = 7$

Step 3 - Find the $y$ -coordinate of the vertex. Substitute the $x$ -coordinate into $f (x)$ . $f (7) = (7 - 4) (7 - 10) = (3) (- 3) = - 9$ The vertex is at $(7, - 9)$ .

2.

What is the axis of symmetry of the quadratic?

Compare your answer:

$x = 7$

The axis of symmetry is the equation where $x$ equals the $x$ -coordinate of the vertex. $x = 7$

7.11.4 Using the Vertex and Axis of Symmetry of Quadratics

Activity

Additional Resources

Axis of Symmetry of a Quadratic

Try It: Axis of Symmetry of a Quadratic