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

$\dfrac{f(b)-f(a)}{b-a}$

Here,

$\dfrac{f(2)-f(-3)}{2-(-3)}=\dfrac{8-(-2)}5=2$

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:

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