Answer :
Sure! Let's work through each part of this problem step by step with the function [tex]\( f(x) = -3x^2 - 12x \)[/tex].
Part (a): Is [tex]\((-2,10)\)[/tex] a point on the graph of [tex]\(f\)[/tex]?
To determine if the point [tex]\((-2, 10)\)[/tex] lies on the graph, substitute [tex]\( x = -2 \)[/tex] into the function and calculate the y-value.
[tex]\[
f(-2) = -3(-2)^2 - 12(-2)
= -3 \times 4 + 24
= -12 + 24
= 12
\][/tex]
The calculated y-value is 12, not 10. Therefore, [tex]\((-2, 10)\)[/tex] is not a point on the graph.
Part (b): Find [tex]\( x \)[/tex] such that [tex]\( (x, 12) \)[/tex] is on the graph of [tex]\(f\)[/tex].
We need to solve the equation [tex]\( f(x) = 12 \)[/tex]:
[tex]\[
-3x^2 - 12x = 12
\][/tex]
Rearranging,
[tex]\[
-3x^2 - 12x - 12 = 0
\][/tex]
To find the solutions, notice that by factoring or using tools:
[tex]\[
x = -2
\][/tex]
Thus, [tex]\( x = -2 \)[/tex] is a solution, meaning the point [tex]\((-2, 12)\)[/tex] is on the graph.
Part (c): Find the y-intercept of the graph of [tex]\(f\)[/tex].
The y-intercept occurs where [tex]\( x = 0 \)[/tex]. Compute [tex]\( f(0) \)[/tex]:
[tex]\[
f(0) = -3(0)^2 - 12(0)
= 0
\][/tex]
So, the y-intercept is at the point [tex]\( (0, 0) \)[/tex].
Part (d): Find all x-intercepts of the graph of [tex]\(f\)[/tex].
The x-intercepts are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
-3x^2 - 12x = 0
\][/tex]
Factor out the greatest common factor:
[tex]\[
-3x(x + 4) = 0
\][/tex]
Solve for [tex]\( x \)[/tex]:
Setting each factor equal to zero gives us:
[tex]\[
x = 0 \quad \text{or} \quad x = -4
\][/tex]
Thus, the x-intercepts are [tex]\( (0, 0) \)[/tex] and [tex]\( (-4, 0) \)[/tex].
This step-by-step explanation accurately concludes the solutions to the given parts of the problem.
Part (a): Is [tex]\((-2,10)\)[/tex] a point on the graph of [tex]\(f\)[/tex]?
To determine if the point [tex]\((-2, 10)\)[/tex] lies on the graph, substitute [tex]\( x = -2 \)[/tex] into the function and calculate the y-value.
[tex]\[
f(-2) = -3(-2)^2 - 12(-2)
= -3 \times 4 + 24
= -12 + 24
= 12
\][/tex]
The calculated y-value is 12, not 10. Therefore, [tex]\((-2, 10)\)[/tex] is not a point on the graph.
Part (b): Find [tex]\( x \)[/tex] such that [tex]\( (x, 12) \)[/tex] is on the graph of [tex]\(f\)[/tex].
We need to solve the equation [tex]\( f(x) = 12 \)[/tex]:
[tex]\[
-3x^2 - 12x = 12
\][/tex]
Rearranging,
[tex]\[
-3x^2 - 12x - 12 = 0
\][/tex]
To find the solutions, notice that by factoring or using tools:
[tex]\[
x = -2
\][/tex]
Thus, [tex]\( x = -2 \)[/tex] is a solution, meaning the point [tex]\((-2, 12)\)[/tex] is on the graph.
Part (c): Find the y-intercept of the graph of [tex]\(f\)[/tex].
The y-intercept occurs where [tex]\( x = 0 \)[/tex]. Compute [tex]\( f(0) \)[/tex]:
[tex]\[
f(0) = -3(0)^2 - 12(0)
= 0
\][/tex]
So, the y-intercept is at the point [tex]\( (0, 0) \)[/tex].
Part (d): Find all x-intercepts of the graph of [tex]\(f\)[/tex].
The x-intercepts are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
-3x^2 - 12x = 0
\][/tex]
Factor out the greatest common factor:
[tex]\[
-3x(x + 4) = 0
\][/tex]
Solve for [tex]\( x \)[/tex]:
Setting each factor equal to zero gives us:
[tex]\[
x = 0 \quad \text{or} \quad x = -4
\][/tex]
Thus, the x-intercepts are [tex]\( (0, 0) \)[/tex] and [tex]\( (-4, 0) \)[/tex].
This step-by-step explanation accurately concludes the solutions to the given parts of the problem.
