Question

given the following function using the techniques of shifting, stretching, reflecting. start with the graph of the basic function ( y = x^3 ), and show all stages. be sure to identify at least three key points. find the domain and the range of the function. ( f(x) = (x + 2)^3 + 5 ) which transformations are needed to graph the function ( f(x) = (x + 2)^3 + 5 )? choose the correct answer below. a. the graph of ( y = x^3 ) should be horizontally shifted to the left by 2 units and shifted vertically down by 5 units. b. the graph of ( y = x^3 ) should be horizontally shifted to the left by 2 units and shifted vertically up by 5 units. c. the graph of ( y = x^3 ) should be horizontally shifted to the right by 2 units and shifted vertically up by 5 units. d. the graph of ( y = x^3 ) should be horizontally shifted to the right by 2 units and shifted vertically down by 5 units. use the graphing tool to graph the equation. fill in the missing coordinates of the points that lie on the graph of ( y = x^3 ) and the corresponding points that lie on the graph of ( f(x) = (x + 2)^3 + 5 ). points that lie on the graph of ( y = x^3 ) (simplify your answers.) corresponding points that lie on the graph of ( f(x) = (x + 2)^3 + 5 ) (type ordered pairs. simplify your answer.) ( (-1, square) ) ( (0, square) ) ( (1, square) ) the domain of ( f(x) ) is ( square ). (type your answer in interval notation.) the range of ( f(x) ) is ( square ). (type your answer in interval notation.)

Explanation:

Part 1: Transformation Question

Step1: Recall Transformation Rules

For a function \( y = f(x) \), the transformation \( y = f(x + h) + k \) means:

• Horizontal shift: If \( h>0 \), shift left by \( h \) units; if \( h<0 \), shift right by \( |h| \) units.
• Vertical shift: If \( k>0 \), shift up by \( k \) units; if \( k<0 \), shift down by \( |k| \) units.

For \( f(x)=(x + 2)^3+5 \), compared to \( y = x^3 \) (\( f(x)=x^3 \)), we have \( h = 2 \) (so left shift by 2 units) and \( k = 5 \) (so up shift by 5 units).

Step2: Evaluate Options

• Option A: Down by 5? No, \( k = 5>0 \) (up). Eliminate.
• Option B: Left by 2 (\( h = 2>0 \)) and up by 5 (\( k = 5>0 \)). Correct.
• Option C: Right by 2? \( h = 2>0 \) means left, not right. Eliminate.
• Option D: Right and down? Both wrong. Eliminate.

Step1: Find Points on \( y = x^3 \)

For \( y = x^3 \):

• When \( x=-1 \): \( y=(-1)^3=-1 \). So point \( (-1, -1) \).
• When \( x = 0 \): \( y=0^3=0 \). So point \( (0, 0) \).
• When \( x = 1 \): \( y=1^3=1 \). So point \( (1, 1) \).

Step2: Find Corresponding Points on \( f(x)=(x + 2)^3+5 \)

To find the corresponding point, we use the transformation: if \( (x, y) \) is on \( y = x^3 \), then the corresponding point on \( f(x) \) is \( (x - 2, y + 5) \) (since we shifted left by 2 (so \( x_{\text{new}}=x - 2 \)) and up by 5 (so \( y_{\text{new}}=y + 5 \))).

• For \( (-1, -1) \):
• New \( x \): \( -1-2=-3 \)
• New \( y \): \( -1 + 5 = 4 \)
• So point \( (-3, 4) \).

• For \( (0, 0) \):
• New \( x \): \( 0-2=-2 \)
• New \( y \): \( 0 + 5 = 5 \)
• So point \( (-2, 5) \).

• For \( (1, 1) \):
• New \( x \): \( 1-2=-1 \)
• New \( y \): \( 1 + 5 = 6 \)
• So point \( (-1, 6) \).

Step1: Recall Domain of Cubic Function

The function \( f(x)=(x + 2)^3+5 \) is a cubic function. Cubic functions (polynomials of degree 3) have a domain of all real numbers because there are no restrictions (no division by zero, no square roots of negative numbers, etc.).

In interval notation, all real numbers is \( (-\infty, \infty) \).

Answer:

B. The graph of \( y = x^3 \) should be horizontally shifted to the left by 2 units and shifted vertically up by 5 units.

Part 2: Filling Coordinates