To analyze the transformations that have occurred on the parent quadratic function \( f(x) = x^2 \) based on the points you've provided, we need to determine how those points relate to the standard form of a quadratic.
The general form of a quadratic function can be written as: \[ f(x) = a(x - h)^2 + k \] where \( (h, k) \) represents the vertex of the parabola and \( a \) affects the width and direction (upward or downward) of the parabola.
Let's analyze the points you provided:
- \( (0, -3) \)
- \( (-4, -4) \)
- \( (-2, -2) \)
Step 1: Identify transformations.
- The point \( (0, -3) \) suggests that when \( x = 0 \), the function value, or the vertical shift from the origin, is \( -3 \). This suggests a vertical shift downwards by 3 units.
Step 2: Determine the vertex.
To identify the vertex of the parabola, we can observe where the points cluster and their relative positions. The point \( (0, -3) \) may indicate a vertex, but we need further analysis.
By examining the other points:
- \( (-4, -4) \)
- \( (-2, -2) \)
The point \( (-2, -2) \) is \( 2 \) units to the left of \( (0, -3) \) and shows an upward pattern, suggesting the vertex is between \( (0, -3) \) and some point lower than that.
Step 3: Compute symmetry and shape.
For a quadratic function, points generally have a symmetric nature around the vertex.
- If we assume a vertex at \( (-2, -3) \) (the x-coordinates of the given points suggest this), we can start to postulate a function based on possible transformations.
Step 4: Using transformations.
Based on the observations:
- The function appears to not only have a vertical shift downward due to the vertex but also reflects downward, as seen from the negative values.
Thus we can formulate: \[ f(x) = a(x + 2)^2 - 3 \] We must find \( a \) using another point. Using \( (0, -3) \): \[ -3 = a(0 + 2)^2 - 3 \] Thus: \[ 0 = 4a \implies a = 0 \] That was too quick; let’s substitute in an existing known point—e.g., \( (-4, -4) \):
- Plugging gives: \[ -4 = a(-4 + 2)^2 - 3 \] \[ -4 = a(4) - 3 \] \[ 4a = -1 \implies a = -\frac{1}{4} \]
Conclusion
The original parent function \( f(x) = x^2 \) has undergone the following transformations:
- A vertical shift downward by 3 units.
- Reflected over the x-axis due to \( a = -\frac{1}{4} \).
The transformed quadratic function describing the transformation would be: \[ f(x) = -\frac{1}{4}(x + 2)^2 - 3 \]
1 answer