Question
6. Find the axis of symmetry and the \( y \)-intercept \( f \) a. \( y=2 x^{2}+8 x-5 \)
Ask by Rogers Wagner. in the United States
Jan 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
- **Axis of Symmetry:** \( x = -2 \)
- **\( y \)-Intercept:** \( (0, -5) \)
Solution
To analyze the quadratic function \( y = 2x^{2} + 8x - 5 \), we'll determine both the **axis of symmetry** and the **\( y \)-intercept**.
### 1. Axis of Symmetry
For any quadratic function in the form \( y = ax^{2} + bx + c \), the **axis of symmetry** can be found using the formula:
\[
x = -\frac{b}{2a}
\]
**Given:**
- \( a = 2 \)
- \( b = 8 \)
**Calculation:**
\[
x = -\frac{8}{2 \times 2} = -\frac{8}{4} = -2
\]
**Result:**
- **Axis of Symmetry:** \( x = -2 \)
### 2. \( y \)-Intercept
The **\( y \)-intercept** occurs where the graph crosses the \( y \)-axis. At this point, the value of \( x \) is **0**.
**To find the \( y \)-intercept:**
- Substitute \( x = 0 \) into the original equation.
\[
y = 2(0)^{2} + 8(0) - 5 = 0 + 0 - 5 = -5
\]
**Result:**
- **\( y \)-Intercept:** \( (0, -5) \) or simply \( y = -5 \)
### Summary
- **Axis of Symmetry:** \( x = -2 \)
- **\( y \)-Intercept:** \( (0, -5) \)
These properties provide valuable information about the symmetry and position of the quadratic graph on the coordinate plane.
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The axis of symmetry for a quadratic function in the form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). In this case, \( a = 2 \) and \( b = 8 \). Plugging in these values gives \( x = -\frac{8}{2 \cdot 2} = -2 \). So, the axis of symmetry is \( x = -2 \). To find the \( y \)-intercept, substitute \( x = 0 \) into the original equation. Thus, \( y = 2(0)^2 + 8(0) - 5 = -5 \). Therefore, the \( y \)-intercept is at the point \( (0, -5) \).
