Transformations of Function Graphs
Students predict and explain how translations, reflections, and stretches change the graph and equation of a parent function.
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Parent Functions
A parent function is a basic function used as a starting point for a family of related graphs. Common parent functions include the linear function f(x) = x, quadratic function f(x) = x², absolute value function f(x) = |x|, and square root function f(x) = √x. Key features help identify each parent graph, such as its shape, intercepts, vertex, domain, and range. For example, f(x) = x² has a U-shaped graph with vertex (0, 0) and symmetry about the y-axis. Transformations move, flip, stretch, or compress this graph while preserving its basic quadratic shape. Recognizing the parent function makes it easier to predict how changes in an equation affect the graph.
Vertical and Horizontal Shifts
A shift moves a graph without changing its shape or orientation. Replacing f(x) with f(x) + k shifts the graph vertically: upward when k > 0 and downward when k < 0. Replacing f(x) with f(x + k) shifts the graph horizontally in the opposite direction from the sign inside the parentheses. Thus, f(x + 3) moves left 3 units, while f(x - 3) moves right 3 units. For example, starting with f(x) = x², the equation g(x) = (x - 2)² + 3 shifts the parabola right 2 units and up 3 units. Its vertex moves from (0, 0) to (2, 3), but the graph remains the same size and still opens upward.
Reflections
A reflection flips a graph across a line. Replacing f(x) with -f(x) reflects the graph across the x-axis because every output changes sign. Each point (x, y) becomes (x, -y). Replacing f(x) with f(-x) reflects the graph across the y-axis because every input changes sign. Each point (x, y) becomes (-x, y). For example, if f(x) = √x, then y = -√x reflects the curve across the x-axis, while y = √(-x) reflects it across the y-axis. A graph that is already symmetric may look unchanged after a reflection. For instance, y = x² and y = (-x)² are identical because the quadratic parent function is symmetric about the y-axis.
Stretches and Compressions
Multiplying a function outside or inside changes its scale. For y = kf(x), every output is multiplied by k. If |k| > 1, the graph is stretched vertically; if 0 < |k| < 1, it is compressed vertically. For y = f(kx), the horizontal effect is reciprocal: if |k| > 1, the graph is compressed horizontally by a factor of 1/|k|; if 0 < |k| < 1, it is stretched horizontally. For example, compared with f(x) = x², g(x) = 2x² is a vertical stretch by a factor of 2. The point (1, 1) becomes (1, 2). In contrast, h(x) = (2x)² is a horizontal compression by a factor of 1/2, so the point (1, 1) corresponds to (1/2, 1).
Matching Equations to Graphs
To match an equation to a graph, first identify the parent shape, then examine location, orientation, and scale. For a quadratic written as y = a(x - h)² + k, the vertex is (h, k). The sign of a tells whether the graph opens upward or downward, and |a| indicates vertical stretch or compression. Suppose a graph has vertex (-1, 3), opens downward, and passes through (0, 1). Begin with y = a(x + 1)² + 3. Substituting (0, 1) gives 1 = a(1)² + 3, so a = -2. The matching equation is y = -2(x + 1)² + 3. Checking several key points helps confirm a match and prevents confusion about horizontal signs or scale factors.