Question

Consider the function y= x^2 +4x-4.

a) what is the vertex of this function? Show your work.
b) what is the equation of the axis of symmetry? Explain how you know.
c) What is the y- intercept?
d) Graph the line of symmetry. Plot the vertex and the point containing the y-intercept. Then plot another point on the graph and use the plotted points and the axis of symmetry to plot two more points. Draw the graph of the function through the points.

Answer 1

Part a) The vertex is the point (-2,-8)

Part b) The equation of the axis of symmetry is x=-2

Part c) The y-intercept is the point (0,-4)

Part d) The graph in the attached figure

Explanation:

we know that

The equation of a vertical parabola in vertex form is equal to

`y=a(x-h)^(2) +k`

where

(h,k) is the vertex of the parabola

if a> 0 then the parabola open upward (vertex is a minimum)

if a<0 then the parabola open downward (vertex is a maximum)

The axis of symmetry of a vertical parabola is equal to the x-coordinate of the vertex

so

x=h ----> equation of the axis of symmetry

In this problem we have

`y=x^(2)+4x-4`

This is the equation of a vertical parabola open upward

The vertex is a minimum

Part a)

what is the vertex of this function?

Convert the function into vertex form

`y=x^(2)+4x-4`

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`y+4=x^(2)+4x`

Complete the square. Remember to balance the equation by adding the same constants to each side.

`y+4+4=(x^(2)+4x+4)`

`y+8=(x^(2)+4x+4)`

Rewrite as perfect squares

`y+8=(x+2)^(2)`

`y=(x+2)^(2)-8` ----> equation in vertex form

The vertex is the point (-2,-8)

Part b) what is the equation of the axis of symmetry?

we know that

The axis of symmetry of a vertical parabola is equal to the x-coordinate of the vertex

so

x=h ----> equation of the axis of symmetry

The vertex is the point (-2,-8)

The x-coordinate of the vertex is -2

therefore

The equation of the axis of symmetry is x=-2

Part c) What is the y- intercept?

we know that

The y-intercept is the value of y when the value of x is equal to zero

so

For x=0

`y=(0)^(2)+4(0)-4`

`y=-4`

The y-intercept is the point (0,-4)

Part d) Graph the line of symmetry. Plot the vertex and the point containing the y-intercept. Then plot another point on the graph and use the plotted points and the axis of symmetry to plot two more points. Draw the graph of the function through the points

To plot the function find the x-intercepts

we know that

The x-intercept is the value of x when the value of y is equal to zero

For y=0

`0=(x+2)^(2)-8`

`(x+2)^(2)=8`

square root both sides

`x+2=(+/-)2√(2)`

`x=-2(+/-)2√(2)`

`x1=-2(+)2√(2)=0.828`

`x2=-2(-)2√(2)=-4.828`

the graph in the attached figure