Question

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1. which equation matches the graph? circle the correct answer.

a.) ( g(x) = (x + 3)^3 + 2 )
b.) ( g(x) = (x - 3)^3 + 2 )
c.) ( g(x) = -(x - 2)^3 + 3 )
d.) ( g(x) = (x + 2)^3 - 3 )

1. consider the function ( f(x) = (x + 3)^2 ). select true or false for each statement about a graph of the function.

a.) it crosses the x-axis only one time. (__) true (__) false
b.) it is tangent to (touches) the x-axis at ( x = 0 ). (__) true (__) false
c.) it has an x-intercept at ( x = -3 ). (__) true (__) false
d.) it has 2 turning points. (__) true (__) false

1. write the polynomial ( -23x^7 + x^9 - 6x^3 + 10 + 2x^2 ) in standard form, and then identify the degree and leading coefficient.

standard form: (________________________)
degree: (______) leading coefficient: (______)

1. add the polynomials.

( (x^7 + 3x^6 - x^2 + 4) + (3x^7 - x^4 + 3x^2 - 9) )

Explanation:

Question 5

Step1: Analyze the graph's shape and transformations

The graph of a cubic function \( y = a(x - h)^3 + k \) has a point of inflection at \( (h, k) \). The given graph is a cubic with a negative leading coefficient (since it falls to the right) and the inflection point seems to be around \( (2, 3) \). Let's check each option:

• Option A: \( g(x)=(x + 3)^3+2 \) has a positive leading coefficient (opens up on the right), so incorrect.
• Option B: \( g(x)=-(x - 3)^3+2 \) has inflection point \( (3, 2) \), not matching the graph.
• Option C: \( g(x)=-(x - 2)^3+3 \) has a negative leading coefficient and inflection point \( (2, 3) \), matches the shape and position.
• Option D: \( g(x)=(x + 2)^3-3 \) has positive leading coefficient and inflection point \( (-2, -3) \), incorrect.

Step2: Confirm the correct option

After analyzing the leading coefficient and inflection point, Option C is correct.

To find x - intercepts, set \( f(x)=x^3(x + 3)=0 \). Solutions are \( x = 0 \) (with multiplicity 3) and \( x=-3 \). So it crosses the x - axis at \( x = 0 \) (since multiplicity is odd, it crosses) and \( x=-3 \), so it crosses the x - axis two times. So the statement "It crosses the x - axis only one time" is False.

A function is tangent to the x - axis at a root when the multiplicity of the root is even. For \( x = 0 \), the multiplicity is 3 (odd), so the graph crosses the x - axis at \( x = 0 \), not tangent. So the statement "It is tangent to (touches) the x - axis at \( x = 0 \)" is False.

Answer:

C. \( g(x) = -(x - 2)^3 + 3 \)

Question 6

Part A