Question

100 points!!

1. What is the average rate of change of the quadratic function from
x = -3 to x = 2? f(x)=x^2+3x−2
Just type in the numeric answer.


2. What is the vertex of the graph of

f(x)=x^2+10x−9?

3. How many solutions does the following quadratic function have?

−8x^2+2x−7=0


4. Find the discriminant.

6q^2+8q−3=0

Answer 1

1. The average rate of change of a function
`f(x)` over an interval
`[a,b]` is given by the difference quotient

`(f(b)-f(a))/(b-a)`

Here,

`(f(2)-f(-3))/(2-(-3))=\frac{8-(-2)}5=2`

2. Complete the square to rewrite the quadratic in vertex form:

`x^2+10x-9=x^2+10x+25-34=(x+5)^2-34`

which indicates its vertex occurs at the point (-5, -34).

3. Check the discriminant. For a quadratic polynomial
`P_2(x)=ax^2+bx+c`, the discriminant is

`\Delta_(P_2)=b^2-4ac`

`\implies\Delta=2^2-4(-8)(-7)=-220<0`

Because the discriminant is negative, there are two complex roots.

4. Same as before:

`\Delta=8^2-4(6)(-3)=136`