Question

question 10 (multiple choice worth 6 points) (06.04r mc)
mary is analyzing a quadratic function f(x) and a linear function g(x). will they intersect?
f(x)
g(x)

x	g(x)
1	-1
2	0
3	1

yes, at positive x-coordinates
yes, at negative x-coordinates
yes, at negative and positive x-coordinates
no, they will not intersect

Explanation:

Step1: Find quadratic function f(x)

The parabola has vertex $(0,0)$ and passes through $(2,4)$. Use vertex form $f(x)=ax^2$. Substitute $(2,4)$:
$4=a(2)^2 \implies 4=4a \implies a=1$. So $f(x)=x^2$.

Step2: Find linear function g(x)

Use points $(1,-1)$ and $(2,0)$. Slope $m=\frac{0-(-1)}{2-1}=1$. Use point-slope form $y-y_1=m(x-x_1)$ with $(2,0)$:
$g(x)-0=1(x-2) \implies g(x)=x-2$.

Step3: Find intersection points

Set $f(x)=g(x)$:
$x^2=x-2 \implies x^2 - x + 2=0$.

Step4: Check discriminant

Discriminant $D=b^2-4ac$ for $ax^2+bx+c=0$:
$D=(-1)^2-4(1)(2)=1-8=-7$.

Step5: Interpret discriminant

Since $D<0$, no real solutions exist.

Answer:

No, they will not intersect