Question

Graph and compare the shapes and key features of the functions f(x)=(x+2)^2 and g(x)=2x+4. Which of the following statements is true?
Responses

f(x) is a straight line and g(x) is a u-shaped curve

both functions have a minimum value of (-2, 0)

f(x) is a u-shaped curve and g(x) is a straight line

both functions have a x-intercept of (4, 0)

Answer 1

To analyze and compare the functions \( f(x) = (x + 2)^2 \) and \( g(x) = 2x + 4 \), we can look at their shapes and key features.

### Function Analysis

1. **For \( f(x) = (x + 2)^2 \)**:
- **Shape**: This is a quadratic function, which means it is a parabola.
- **Direction**: The parabola opens upwards because the coefficient of the squared term is positive.
- **Vertex**: The vertex of the parabola is at the point \((-2, 0)\), which gives the minimum value of the function.
- **x-intercept**: Set \( f(x) = 0 \) to find the x-intercept:
\[
(x + 2)^2 = 0 \implies x + 2 = 0 \implies x = -2 \quad \text{(Point: } (-2, 0) \text{)}
\]

2. **For \( g(x) = 2x + 4 \)**:
- **Shape**: This is a linear function, which means it is a straight line.
- **Slope**: The slope is 2, indicating the line rises steeply as \( x \) increases.
- **y-intercept**: To find where it crosses the y-axis, when \( x = 0 \):
\[
g(0) = 2(0) + 4 = 4 \quad \text{(Point: } (0, 4) \text{)}
\]
- **x-intercept**: Set \( g(x) = 0 \) to find the x-intercept:
\[
2x + 4 = 0 \implies 2x = -4 \implies x = -2 \quad \text{(Point: } (-2, 0) \text{)}
\]

### Summary of Key Features
- \( f(x) \) is a u-shaped curve (parabola) with a minimum at \((-2, 0)\) and x-intercept at \((-2, 0)\).
- \( g(x) \) is a straight line with an x-intercept at \((-2, 0)\) and a y-intercept at \((0, 4)\).

### Evaluating the Statements
1. **f(x) is a straight line and g(x) is a u-shaped curve**: **False** (f(x) is a parabola, g(x) is a line).
2. **Both functions have a minimum value of (-2, 0)**: **False** (only f(x) has a minimum at this point; g(x) goes to infinity).
3. **f(x) is a u-shaped curve and g(x) is a straight line**: **True** (f(x) is a parabola, g(x) is a line).
4. **Both functions have an x-intercept of (4, 0)**: **False** (Both have x-intercept at (-2, 0)).

### Conclusion
The true statement is:
**f(x) is a u-shaped curve and g(x) is a straight line.**