Shifting Graphs

The accompanying figure shows the graph of y = −x² shifted to four new positions. Write an equation for each new graph.

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Identify the original function: The original function given is y = -x². This is a downward-opening parabola centered at the origin (0,0).

Understand the types of shifts: Graphs can be shifted vertically, horizontally, or both. A vertical shift involves adding or subtracting a constant to the function, while a horizontal shift involves adding or subtracting a constant inside the function's argument.

Determine the vertical shifts: If the graph is moved up or down, this is a vertical shift. For example, if the graph is moved up by 'k' units, the new equation becomes y = -x² + k. If moved down by 'k' units, it becomes y = -x² - k.

Determine the horizontal shifts: If the graph is moved left or right, this is a horizontal shift. For example, if the graph is moved right by 'h' units, the new equation becomes y = -(x-h)². If moved left by 'h' units, it becomes y = -(x+h)².

Combine shifts if necessary: If the graph is shifted both horizontally and vertically, combine the transformations. For example, if the graph is shifted right by 'h' units and up by 'k' units, the equation becomes y = -(x-h)² + k.

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Graph Shifting

Graph shifting involves translating a function's graph horizontally or vertically without altering its shape. A horizontal shift is achieved by adding or subtracting a value from the input variable (x), while a vertical shift is done by adding or subtracting a value from the output variable (y). For example, shifting the graph of y = -x² to the right by 3 units results in the equation y = -(x - 3)².

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Quadratic functions are polynomial functions of degree two, typically expressed in the form y = ax² + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. In the case of y = -x², the parabola opens downwards, and its vertex is at the origin (0,0). Understanding the standard form helps in identifying how shifts affect the graph.

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The vertex form of a quadratic function is given by y = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form is particularly useful for graphing and understanding transformations, as it clearly shows how the graph shifts based on the values of h and k. For instance, if h is positive, the graph shifts to the right, and if k is positive, it shifts upward.

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