Question

the graph of the cube root parent function $y = \sqrt3{x}$ is translated to form $f(x)$ shown on the graph. which equation represents $f(x)$? $\bigcirc\\ f(x) = \sqrt3{x + 6} + 1$ $\bigcirc\\ f(x) = \sqrt3{x - 6} + 1$ $\bigcirc\\ f(x) = \sqrt3{x + 6} - 1$ $\bigcirc\\ f(x) = \sqrt3{x - 6} - 1$

Explanation:

Step1: Recall Cube Root Transformations

The parent function is \( y = \sqrt[3]{x} \). For horizontal shifts: \( y = \sqrt[3]{x - h} \) shifts right \( h \), \( y = \sqrt[3]{x + h} \) shifts left \( h \). Vertical shifts: \( y = \sqrt[3]{x} + k \) shifts up \( k \), \( y = \sqrt[3]{x} - k \) shifts down \( k \).

Step2: Find Key Point on \( f(x) \)

The parent \( y = \sqrt[3]{x} \) has a point at \( (0,0) \). The graph of \( f(x) \) has a key point (where the curve is "centered") at \( (-6, 0) \)? Wait, no—wait, looking at the graph, when \( x = -6 \), what's \( y \)? Wait, no, let's check the y-intercept. The graph crosses the y-axis at \( (0, 3) \)? Wait, no, the grid: when \( x = 0 \), \( y = 3 \)? Wait, no, let's find the vertex-like point. The parent \( y = \sqrt[3]{x} \) passes through \( (0,0) \), \( (1,1) \), \( (-1,-1) \). The transformed graph: let's see the point where the curve is at the "corner"—looking at the graph, when \( x = -6 \), what's \( y \)? Wait, the x-intercept: the graph crosses the x-axis at \( x = -8 \)? No, wait the grid: the x-axis crossing is at \( x = -8 \)? Wait, no, the graph has a point at \( (-8, 0) \)? Wait, no, the leftmost part: when \( x = -8 \), \( y = 0 \)? Wait, let's check the options. Let's test \( x = -6 \) in each option.

Option 1: \( f(-6) = \sqrt[3]{-6 + 6} + 1 = \sqrt[3]{0} + 1 = 0 + 1 = 1 \). No, the graph at \( x = -6 \): wait, maybe better to find the horizontal shift. The parent's "center" is at \( (0,0) \). The transformed graph's center (the point where the curve is symmetric) is at \( (-6, 1) \)? Wait, no—wait, let's take the point where \( x = -6 \), \( y = 1 \)? Wait, no, let's check the y-intercept. For \( x = 0 \):

Option 1: \( f(0) = \sqrt[3]{0 + 6} + 1 = \sqrt[3]{6} + 1 \approx 1.817 + 1 = 2.817 \approx 3 \). Close to the graph's y-intercept (which looks like \( (0, 3) \)). Wait, let's check Option 1: \( f(x) = \sqrt[3]{x + 6} + 1 \). Let's see the horizontal shift: \( x + 6 = 0 \) when \( x = -6 \), so the center is at \( (-6, 1) \). Let's check \( x = -6 \): \( f(-6) = 0 + 1 = 1 \). Then, when \( x = -6 + 1 = -5 \), \( f(-5) = \sqrt[3]{-5 + 6} + 1 = \sqrt[3]{1} + 1 = 1 + 1 = 2 \). When \( x = -6 - 1 = -7 \), \( f(-7) = \sqrt[3]{-7 + 6} + 1 = \sqrt[3]{-1} + 1 = -1 + 1 = 0 \). Wait, that matches the x-intercept at \( x = -7 \)? Wait, no, the graph's x-intercept is at \( x = -8 \)? Wait, maybe I misread. Wait the graph: the x-axis crossing is at \( x = -8 \)? Let's test \( x = -8 \) in Option 1: \( f(-8) = \sqrt[3]{-8 + 6} + 1 = \sqrt[3]{-2} + 1 \approx -1.26 + 1 = -0.26 \), not 0. Option 3: \( f(-8) = \sqrt[3]{-8 + 6} - 1 = \sqrt[3]{-2} - 1 \approx -1.26 - 1 = -2.26 \), no. Option 2: \( f(-8) = \sqrt[3]{-8 - 6} + 1 = \sqrt[3]{-14} + 1 \approx -2.41 + 1 = -1.41 \), no. Option 4: \( f(-8) = \sqrt[3]{-8 - 6} - 1 = \sqrt[3]{-14} - 1 \approx -2.41 - 1 = -3.41 \), no. Wait, maybe the key point is \( (0, 3) \). Let's check \( x = 0 \) in each option:

Option 1: \( \sqrt[3]{6} + 1 \approx 1.817 + 1 = 2.817 \approx 3 \). Good.

Option 2: \( \sqrt[3]{-6} + 1 \approx -1.817 + 1 = -0.817 \), no.

Option 3: \( \sqrt[3]{6} - 1 \approx 1.817 - 1 = 0.817 \), no.

Option 4: \( \sqrt[3]{-6} - 1 \approx -1.817 - 1 = -2.817 \), no.

Also, horizontal shift: parent \( y = \sqrt[3]{x} \) has center at \( (0,0) \). The transformed graph's center (where the curve is "flat") is at \( (-6, 1) \), so horizontal shift left 6 (so \( x + 6 \)) and vertical shift up 1 (so \( +1 \)). Thus, \( f(x) = \sqrt[3]{x + 6} + 1 \).

Answer:

A. \( f(x) = \sqrt[3]{x + 6} + 1 \)