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Find the Vertex Coordinate of $y=-f(x)$ Given Vertex of $y=f(x)$ is (2, -4)
Mathematics
Grade 10 (Junior High School)
Question Content
This is a sketch of the curve with equation $y = f(x)$. It passes through the origin $O$. The only vertex of the curve is at $A(2, -4)$. Write down the coordinates of the vertex of the curve with equation $y = -f(x)$. (1 mark)
Correct Answer
(2, 4)
Detailed Solution Steps
1
Step 1: Recall the transformation rule for $y = -f(x)$: this transformation reflects the graph of $y=f(x)$ across the x-axis.
2
Step 2: For any point $(x, y)$ on $y=f(x)$, the corresponding point on $y=-f(x)$ will have the same x-coordinate, and the y-coordinate will be multiplied by -1.
3
Step 3: Apply the rule to the vertex $A(2, -4)$: keep the x-coordinate 2, multiply the y-coordinate by -1: $-(-4) = 4$.
4
Step 4: The vertex of $y=-f(x)$ is $(2, 4)$.
Knowledge Points Involved
1
Graph Transformation: Reflection over the x-axis ($y=-f(x)$)
The transformation $y=-f(x)$ takes every point $(x, y)$ on the original function $y=f(x)$ and maps it to $(x, -y)$. This flips the entire graph vertically across the x-axis, reversing the sign of all y-values while keeping x-values unchanged. It is used to find the coordinates of points on the transformed function from the original function's points.
2
Vertex of a Quadratic Curve
The vertex is the turning point (minimum or maximum) of a quadratic curve. For a quadratic function, this is the point where the curve changes direction. Transformations of the function will directly affect the coordinates of this vertex following the rules of function transformations.
3
Function Transformation Rules
Function transformations modify the graph of a parent function $y=f(x)$ based on algebraic changes to the function. Rules like $y=-f(x)$ (reflection), $y=f(x)+k$ (vertical shift), $y=f(x-h)$ (horizontal shift) are standard rules used to predict the position of points and the shape of the transformed graph without re-plotting the entire curve.
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