To interpret the key features of the graph of the function \( f(x) = 4x^2 \), we start by comparing it to its parent function, which is \( g(x) = x^2 \), the standard quadratic function.
Key Features:
-
Shape and Direction:
- The graph of \( f(x) = 4x^2 \) retains the U-shape of the parent quadratic function, opening upwards.
-
Vertex:
- The vertex of both \( f(x) \) and \( g(x) \) is at the origin, (0, 0), since there are no horizontal or vertical shifts in \( f(x) \).
-
Width of the Parabola:
- The graph of \( f(x) = 4x^2 \) is narrower than that of \( g(x) = x^2 \). This is because the coefficient 4 (which is greater than 1) causes the parabola to stretch vertically. The larger the coefficient in front of \( x^2 \), the narrower the parabola.
-
Y-intercept:
- Both functions have a y-intercept at \( (0, 0) \).
-
Symmetry:
- The parabola is symmetric with respect to the y-axis, which is a characteristic of all functions of the form \( ax^2 \).
Type of Transformation:
The transformation that occurred from the parent function \( g(x) = x^2 \) to \( f(x) = 4x^2 \) is a vertical stretch:
- Vertical Stretch: The transformation can be described as "stretching" the graph vertically by a factor of 4. For every value of \( x \), the corresponding \( y \)-value of the function \( f(x) \) is quadrupled compared to that of \( g(x) \). Therefore, if you take a point on \( g(x) \), such as \( (1, 1) \), it would correspond to the point \( (1, 4) \) on \( f(x) \).
In summary, the graph of \( f(x) = 4x^2 \) depicts a vertical stretch of the parent function \( g(x) = x^2 \) by a factor of 4.
1 answer