Question
Find the \( x \)-intercepts and \( y \)-intercept of the following function. \[ f(x)=(x-4)(x+1)(x+8) \]
Ask by Chen Bates. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The \( x \)-intercepts are at \( (4, 0) \), \( (-1, 0) \), and \( (-8, 0) \), and the \( y \)-intercept is at \( (0, -32) \).
Solution
To find the \( x \)-intercepts and the \( y \)-intercept of the function \( f(x) = (x-4)(x+1)(x+8) \), follow these steps:
### \( x \)-Intercepts
The \( x \)-intercepts occur where \( f(x) = 0 \). Set the function equal to zero and solve for \( x \):
\[
(x - 4)(x + 1)(x + 8) = 0
\]
This equation is satisfied when any of the factors is zero:
1. \( x - 4 = 0 \) \(\Rightarrow\) \( x = 4 \)
2. \( x + 1 = 0 \) \(\Rightarrow\) \( x = -1 \)
3. \( x + 8 = 0 \) \(\Rightarrow\) \( x = -8 \)
So, the \( x \)-intercepts are:
- \( (4, 0) \)
- \( (-1, 0) \)
- \( (-8, 0) \)
### \( y \)-Intercept
The \( y \)-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the function:
\[
f(0) = (0 - 4)(0 + 1)(0 + 8) = (-4)(1)(8) = -32
\]
So, the \( y \)-intercept is:
- \( (0, -32) \)
### Summary
- **\( x \)-intercepts:** \( (4, 0) \), \( (-1, 0) \), and \( (-8, 0) \)
- **\( y \)-intercept:** \( (0, -32) \)
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution The Deep Dive
To find the \( x \)-intercepts of the function \( f(x)=(x-4)(x+1)(x+8) \), set \( f(x) = 0 \). This gives us: \[ (x-4)(x+1)(x+8) = 0 \] The \( x \)-intercepts occur when any of the factors equal zero: 1. \( x - 4 = 0 \) → \( x = 4 \) 2. \( x + 1 = 0 \) → \( x = -1 \) 3. \( x + 8 = 0 \) → \( x = -8 \) Thus, the \( x \)-intercepts are \( (4, 0) \), \( (-1, 0) \), and \( (-8, 0) \). To find the \( y \)-intercept, evaluate \( f(0) \): \[ f(0) = (0-4)(0+1)(0+8) = (-4)(1)(8) = -32 \] So, the \( y \)-intercept is \( (0, -32) \). In summary, the intercepts are: - \( x \)-intercepts: \( (4, 0) \), \( (-1, 0) \), \( (-8, 0) \) - \( y \)-intercept: \( (0, -32) \)
