Vaughn Daniel
08/08/2024 · Middle School
3. Determine the \( x \)-and \( y \)-intercepts of the graph of the equation. \( \quad y=(x-1)^{2}-4 \)
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Tutor-Verified Answer
Step-by-step Solution
To find the \( x \)- and \( y \)-intercepts of the equation \( y = (x-1)^2 - 4 \), we will follow these steps:
### Finding the \( y \)-intercept:
The \( y \)-intercept occurs when \( x = 0 \). We substitute \( x = 0 \) into the equation:
\[
y = (0 - 1)^2 - 4
\]
\[
y = (-1)^2 - 4
\]
\[
y = 1 - 4
\]
\[
y = -3
\]
Thus, the \( y \)-intercept is at the point \( (0, -3) \).
### Finding the \( x \)-intercepts:
The \( x \)-intercepts occur when \( y = 0 \). We set the equation equal to zero and solve for \( x \):
\[
0 = (x - 1)^2 - 4
\]
\[
(x - 1)^2 = 4
\]
Next, we take the square root of both sides:
\[
x - 1 = 2 \quad \text{or} \quad x - 1 = -2
\]
Solving these equations gives:
1. \( x - 1 = 2 \)
\[
x = 3
\]
2. \( x - 1 = -2 \)
\[
x = -1
\]
Thus, the \( x \)-intercepts are at the points \( (3, 0) \) and \( (-1, 0) \).
### Summary:
- The \( y \)-intercept is \( (0, -3) \).
- The \( x \)-intercepts are \( (3, 0) \) and \( (-1, 0) \).
Quick Answer
The \( y \)-intercept is \( (0, -3) \) and the \( x \)-intercepts are \( (3, 0) \) and \( (-1, 0) \).
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