To compare the graph of a new quadratic function to the parent quadratic function, which is typically represented as \( f(x) = x^2 \), you can focus on several key features. Here's what to examine:
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Vertex: Identify the vertex of the new quadratic function. The parent function has its vertex at the origin (0,0). Check how the vertex has shifted vertically or horizontally in the new function. The vertex form of a quadratic is given by \( f(x) = a(x - h)^2 + k \), where \((h, k)\) represents the vertex.
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Axis of Symmetry: The parent function has a vertical axis of symmetry at \( x = 0 \). For the new function, determine the axis of symmetry, which can be derived from the vertex. This shows whether the graph is shifted left or right.
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Direction of Opening: The parent quadratic opens upwards. If the coefficient \( a \) in the new function (in the standard form \( ax^2 + bx + c \)) is positive, it opens upwards; if negative, it opens downwards. This will affect the overall shape of the graph.
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Width of the Parabola: The value of \( a \) also affects the width of the parabola. If \( |a| > 1 \), the parabola is narrower than the parent function; if \( |a| < 1 \), it is wider.
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Y-Intercept: Check the y-intercept of the new function. The parent function has a y-intercept at (0,0). The new function's y-intercept can be found by evaluating \( f(0) \) and comparing it to the parent function.
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X-Intercepts (Roots): Determine the x-intercepts of the new function by setting \( f(x) = 0 \) and solving for \( x \). Compare these to the x-intercepts of the parent function, which are also at \( x = 0 \).
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End Behavior: Compare how the ends of the new quadratic behave as \( x \) approaches positive and negative infinity. For the parent function, both ends rise to infinity. Check if the new function behaves similarly based on the sign of \( a \).
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Shape and Curvature: Observe the overall shape and curvature of the parabola compared to the parent quadratic function. Note any transformations such as reflections or vertical/horizontal stretches or compressions.
By considering these features, you can effectively compare the graph of the new quadratic function to the parent quadratic function.