Question
Use the calculator to evaluate the function y = one-quarter x2 – 2 for the given values of x.
x = –4, y =
x = 6, y =
x = –4, y =
x = 6, y =
Answers
GPT-4o mini
To evaluate the function \( y = \frac{1}{4}x^2 - 2 \) for the given values of \( x \):
1. **For \( x = -4 \)**:
\[
y = \frac{1}{4}(-4)^2 - 2
\]
\[
= \frac{1}{4}(16) - 2
\]
\[
= 4 - 2
\]
\[
= 2
\]
So, when \( x = -4 \), \( y = 2 \).
2. **For \( x = 6 \)**:
\[
y = \frac{1}{4}(6)^2 - 2
\]
\[
= \frac{1}{4}(36) - 2
\]
\[
= 9 - 2
\]
\[
= 7
\]
So, when \( x = 6 \), \( y = 7 \).
### Summary:
- For \( x = -4, y = 2 \)
- For \( x = 6, y = 7 \)
1. **For \( x = -4 \)**:
\[
y = \frac{1}{4}(-4)^2 - 2
\]
\[
= \frac{1}{4}(16) - 2
\]
\[
= 4 - 2
\]
\[
= 2
\]
So, when \( x = -4 \), \( y = 2 \).
2. **For \( x = 6 \)**:
\[
y = \frac{1}{4}(6)^2 - 2
\]
\[
= \frac{1}{4}(36) - 2
\]
\[
= 9 - 2
\]
\[
= 7
\]
So, when \( x = 6 \), \( y = 7 \).
### Summary:
- For \( x = -4, y = 2 \)
- For \( x = 6, y = 7 \)
Graph the function y = x2 – 2. Which key features describe the function? Check all that apply.
The point (1.414, 0) is an x-intercept.
The point (–2, 0) is the y-intercept.
The point (0, –2) is the maximum.
The function is decreasing over the interval (–infinity, 0].
The function is increasing over the interval (–infinity, infinity).
The point (1.414, 0) is an x-intercept.
The point (–2, 0) is the y-intercept.
The point (0, –2) is the maximum.
The function is decreasing over the interval (–infinity, 0].
The function is increasing over the interval (–infinity, infinity).
GPT-4o mini
Let's analyze the function \( y = x^2 - 2 \) to identify its key features:
1. **Intercepts**:
- **X-intercepts**: To find the x-intercepts, we set \( y = 0 \):
\[
0 = x^2 - 2 \implies x^2 = 2 \implies x = \pm\sqrt{2} \approx \pm 1.414
\]
So the x-intercepts are \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \), or approximately \( (1.414, 0) \) and \( (-1.414, 0) \). Thus, the statement **"The point (1.414, 0) is an x-intercept."** is **true**.
- **Y-intercept**: To find the y-intercept, we set \( x = 0 \):
\[
y = 0^2 - 2 = -2
\]
So the y-intercept is \( (0, -2) \). Thus, the statement **"The point (–2, 0) is the y-intercept."** is **false** (the correct y-intercept is \( (0, -2) \)).
2. **Maximum/Minimum**:
- The function \( y = x^2 - 2 \) is a parabola that opens upwards (since the coefficient of \( x^2 \) is positive), which means it has a minimum point, not a maximum. The vertex of the parabola (where the minimum occurs) can be found at:
\[
y = -2 \quad \text{(at } x = 0 \text{)}
\]
Therefore, the statement **"The point (0, –2) is the maximum."** is **false** (it is the minimum).
3. **Increasing/Decreasing Intervals**:
- The function decreases on the interval \( (-\infty, 0] \) and increases on the interval \( [0, \infty) \).
Thus, the statement **"The function is decreasing over the interval (–infinity, 0]."** is **true**.
- The statement **"The function is increasing over the interval (–infinity, infinity)."** is **false** (it only increases from \( [0, \infty) \)).
### Summary of Key Features:
- The point \( (1.414, 0) \) is an x-intercept: **True**
- The point \( (–2, 0) \) is the y-intercept: **False** (it should be \( (0, -2) \))
- The point \( (0, –2) \) is the maximum: **False** (it is the minimum)
- The function is decreasing over the interval \( (–\infty, 0] \): **True**
- The function is increasing over the interval \( (–\infty, \infty) \): **False** (it only increases on \( [0, \infty) \))
**Final True Statements**:
1. The point \( (1.414, 0) \) is an x-intercept.
2. The function is decreasing over the interval \( (–\infty, 0] \).
1. **Intercepts**:
- **X-intercepts**: To find the x-intercepts, we set \( y = 0 \):
\[
0 = x^2 - 2 \implies x^2 = 2 \implies x = \pm\sqrt{2} \approx \pm 1.414
\]
So the x-intercepts are \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \), or approximately \( (1.414, 0) \) and \( (-1.414, 0) \). Thus, the statement **"The point (1.414, 0) is an x-intercept."** is **true**.
- **Y-intercept**: To find the y-intercept, we set \( x = 0 \):
\[
y = 0^2 - 2 = -2
\]
So the y-intercept is \( (0, -2) \). Thus, the statement **"The point (–2, 0) is the y-intercept."** is **false** (the correct y-intercept is \( (0, -2) \)).
2. **Maximum/Minimum**:
- The function \( y = x^2 - 2 \) is a parabola that opens upwards (since the coefficient of \( x^2 \) is positive), which means it has a minimum point, not a maximum. The vertex of the parabola (where the minimum occurs) can be found at:
\[
y = -2 \quad \text{(at } x = 0 \text{)}
\]
Therefore, the statement **"The point (0, –2) is the maximum."** is **false** (it is the minimum).
3. **Increasing/Decreasing Intervals**:
- The function decreases on the interval \( (-\infty, 0] \) and increases on the interval \( [0, \infty) \).
Thus, the statement **"The function is decreasing over the interval (–infinity, 0]."** is **true**.
- The statement **"The function is increasing over the interval (–infinity, infinity)."** is **false** (it only increases from \( [0, \infty) \)).
### Summary of Key Features:
- The point \( (1.414, 0) \) is an x-intercept: **True**
- The point \( (–2, 0) \) is the y-intercept: **False** (it should be \( (0, -2) \))
- The point \( (0, –2) \) is the maximum: **False** (it is the minimum)
- The function is decreasing over the interval \( (–\infty, 0] \): **True**
- The function is increasing over the interval \( (–\infty, \infty) \): **False** (it only increases on \( [0, \infty) \))
**Final True Statements**:
1. The point \( (1.414, 0) \) is an x-intercept.
2. The function is decreasing over the interval \( (–\infty, 0] \).