Answer:
Part 1) The vertex is the point (4,4)
Part 2) The x-intercepts are the points (0,0) and (8,0)
Part 3) The y-intercept is the point (0,0)
Part 4) The equation of the line of symmetry is x=4
Part 5) The graph in the attached figure
Step-by-step explanation:
The correct quadratic equation is
[tex]y= \frac{1}{4}(8x-x^{2})[/tex]
Part 1) Find the coordinates of the vertex
we have
[tex]y= \frac{1}{4}(8x-x^{2})[/tex]
This is a vertical parabola open down (leading coefficient is negative)
The vertex is a maximum
Convert to vertex form
Factor -1
[tex]y= -\frac{1}{4}(x^{2}-8x)[/tex]
Complete the square
[tex]y= -\frac{1}{4}(x^{2}-8x+16))+4[/tex]
Rewrite as perfect squares
[tex]y= -\frac{1}{4}(x-4)^{2}+4[/tex] ----> equation in vertex form
The vertex is the point (4,4)
Part 2) Find the x-intercepts
Remember that
The x-intercepts are the values of x when the value of y is equal to zero
so
For y=0
[tex]-\frac{1}{4}(x-4)^{2}+4=0[/tex]
solve for x
Multiply both sides by 4
[tex](x-4)^{2}=16[/tex]
square root both sides
[tex]x-4=\pm4\\x=4\pm4\\x=4+4=8\\x=4-4=0[/tex]
therefore
The x-intercepts are the points (0,0) and (8,0)
Part 3) Find the y-intercept
Remember that
The y-intercept is the value of y when the value of x is equal to zero
so
For x=0
[tex]y= -\frac{1}{4}(0-4)^{2}+4[/tex]
[tex]y= -4+4=0[/tex]
therefore
The y-intercept is the point (0,0)
Part 4)
Find the equation of the line of symmetry
we know that
In a vertical parabola the equation of the line of symmetry is equal to the x-coordinate of the vertex
The vertex is the point (4,4)
so
The equation of the line of symmetry is x=4
Part 5) Graph the function
we have
The vertex is the point (4,4)
The x-intercepts are the points (0,0) and (8,0)
The y-intercept is the point (0,0)
