PLEASEE HELPP ASAPPP QUADRATIC FUNCTIONS

Question

For each function, find the x intercepts and then determine the coordinates of the vertex and the equation of the axis of symmetry.

1) -(m-5)(m+7)=0

2) (p-8)(p+1)=0

Make sure work is shown to find x intercepts and vertex and both are given as coordinates and an equation for AOS.

Answer 1

Since the quadratic equations are in factored form already, we can easily find the x-intercepts and, from those, the vertex and AOS.

Starting with the first problem, the values of "m" that make each factor equal to zero are the x-intercepts. So setting each individual factor equal to 0, we have: 1) m - 5 = 0, so m = 5; 2) m + 7 = 0, so m = -7. So the x-intercepts are x = 5 and x = -7 ( or, ( 5 , 0) and ( -7, 0 ))

Now, due to the symmetry of the graphs quadratic functions ( called " parabolas" ), the vertex will have an x-coordinate that is exactly halfway between the x-intercepts. So halfway between 5 and -7 is -1 so the x-coordinate of the vertex is -1. To find the y-coordinate, we simply plug in the x value of -1 into the original equation, except that we set it equal to y instead of 0, and we determine the y-coordinate. ** Note: the minus sign in front of the (m-5) factor indicates this parabola opens down, so the vertex will be above the x-axis and thus the y-coordinate should be a positive value. So it will look like this:

-( -1 - 5)( -1 + 7 ) = y. Doing the arithmetic, we have y = 36, so the vertex is at ( -1, 36 ). Since the axis of symmetry splits the parabola perfectly in half, it must pass through the vertex and its equation, as a vertical line, will be x = the x-coordinate of the vertex, so the AOS has the equation x = -1.

For the second problem, it can be done exactly the same way, except that you should note that this time, with no minus sign in front of the first factor in the equation, this parabola will open up and so your vertex will be below the x-intercepts. Good luck!

Answer 2

You can set up both functions in terms of x and y

1) -(m-5)(m+7)

Let x = m

-(x-5)(x+7)=0

To get the x intercepts just set each factor equal to zero and solve x

Be careful with the sign multiply the factors (FOIL) to get the quadratic

Set y equal to your quadratic

Then use x = -b/2a this is your AOS and also the x coordinate of the vertex

Plug x into your quadratic to get the y coordinate of the vertex

Number 2 is done below

2) (p-8)(p+1)

Let x = p below

(x-8)(x+1)=0

Set each factor equal to 0 to find the x intercepts

x-8=0

x=8 (8,0) is one of the x intercepts

x+1=0

x=-1 (-1,0) is the other x intercept

Set y = the product of the factors (FOIL), the quadratic

y = x² -7x-8

a = 1

b = -7

-b/2a = the x coordinate of the vertex

x= -b/2a= -(-7/2) = 3.5

x = 3.5 is your AOS

Plug this in for x and solve for y to get the y coordinate of the vertex

y=(3.5)² - 7(3.5) -8

y= -20.25

Vertex is (3.5, -20.25)

Give it a try for 1, be careful with the sign when you do 1.

-(x -5)(x+7)

and

(-x+5)(x+7)

Are identical

You can check your intercepts and vertex at Desmos.com