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
-
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: \[ (x + 2)^2 = 0 \implies x + 2 = 0 \implies x = -2 \quad \text{(Point: } (-2, 0) \text{)} \]
-
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
- 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).
- Both functions have a minimum value of (-2, 0): False (only f(x) has a minimum at this point; g(x) goes to infinity).
- 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).
- 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.