Question

a quadratic function $f(x)$ is hidden from view. you must find the $y$-intercept(s) of $f(x)$ and write the answer(s) in the form $(x, y)$. choose the form of the quadratic function $f(x)$ that you would like to see in order to answer the question most efficiently.

form: vertex form $f(x) = 3(x + 2)^2 - 48$

answer attempt 1 out of 2
there are no $y$-intercepts
there are no $y$-intercepts
there is one $y$-intercept
there are two $y$-intercepts

Explanation:

Step1: Recall y-intercept definition

The y-intercept of a function \( f(x) \) is the point where \( x = 0 \), so we substitute \( x = 0 \) into the function.

Step2: Substitute \( x = 0 \) into \( f(x) = 3(x + 2)^2 - 48 \)

First, calculate \( (0 + 2)^2 = 2^2 = 4 \). Then, \( 3\times4 = 12 \). Finally, \( 12 - 48 = -36 \). So when \( x = 0 \), \( y = -36 \), which means there is one y-intercept at \( (0, -36) \). But wait, the dropdown says "There are no y-intercepts" which is wrong. Wait, no—wait, the process: y-intercept is when x=0, so let's do it again. \( f(0)=3(0 + 2)^2 - 48 = 3\times4 - 48 = 12 - 48 = -36 \). So the y-intercept is (0, -36), so there is one y-intercept. Wait, the initial dropdown was wrong. So the correct answer is that there is one y-intercept, at (0, -36). But the question is to find the y-intercept(s). Wait, the problem says "find the y-intercept(s)". So step by step:

1. To find the y-intercept, set \( x = 0 \) in the function \( f(x) = 3(x + 2)^2 - 48 \).
2. Substitute \( x = 0 \): \( f(0) = 3(0 + 2)^2 - 48 \).
3. Calculate \( (0 + 2)^2 = 4 \), then \( 3\times4 = 12 \), then \( 12 - 48 = -36 \).
4. So the y-intercept is at \( (0, -36) \), so there is one y-intercept.

Wait, but the dropdown in the image had "There are no y-intercepts" selected, which is incorrect. So the correct process is to substitute x=0, find y, and that's the y-intercept. Since a quadratic function is a parabola, and all parabolas (except maybe if it's a vertical line, but quadratics are functions, so vertical line test passes) have exactly one y-intercept (since x=0 is a single value, and the function is defined for x=0 here). So the y-intercept is (0, -36), so there is one y-intercept.

Answer:

There is one y - intercept, and the y - intercept is \((0, -36)\)