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Algebra 1

7.11.4 Using the Vertex and Axis of Symmetry of Quadratics

Algebra 17.11.4 Using the Vertex and Axis of Symmetry of Quadratics

7.11.4 Using the Vertex and Axis of Symmetry of Quadratics

7.11.4 • Using the Vertex and Axis of Symmetry of Quadratics

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 function f ( x ) = ( x − 2 ) ( x + 4 ) . f ( x ) = ( x − 2 ) ( x + 4 ) .

1.

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

2.

Find the coordinate of the vertex.

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.

4.

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

5.

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

6.

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

7.

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

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

Self Check

What is the equation of the axis of symmetry for f ( x ) = ( x + 1 ) ( x − 3 ) ?
  1. x = − 4
  2. x = 3
  3. x = − 1
  4. 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 x -coordinate of the vertex.

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

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

Example

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

Step 1 -  Find the x x -intercepts.

Set each factor equal to 0.

x + 2 = 0 x + 2 = 0

x = − 2 x = − 2

x − 6 = 0 x − 6 = 0

x = 6 x = 6

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

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

x = 6 + ( − 2 ) 2 = 2 x = 6 + ( − 2 ) 2 = 2

x = 6 + ( − 2 ) 2 = 2 x = 6 + ( − 2 ) 2 = 2

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

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

f ( 2 ) = ( 2 + 2 ) ( 2 − 6 ) = ( 4 ) ( − 4 ) = − 6 f ( 2 ) = ( 2 + 2 ) ( 2 − 6 ) = ( 4 ) ( − 4 ) = − 6

The vertex is at ( 2 , − 16 ) ( 2 , − 16 ) .

Step 4 - Find the axis of symmetry.

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

x = 2 x = 2

Try it

Axis of Symmetry of a Quadratic

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

1.

What is the vertex of the quadratic?

2.

What is the axis of symmetry of the quadratic?

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