The three types of transformations of a graph are stretches, reflections and shifts. The vertical stretch of a graph measures the stretching or shrinking factor in the vertical direction. For example, if a function increases three times as fast as its parent function, it has a stretch factor of 3. To find the vertical stretch of a graph, create a function based on its transformation from the parent function, plug in an (x, y) pair from the graph and solve for the value A of the stretch.

Identify the type of function in the graph as a quadratic, cubic, trigonometric or exponential function based on such features as its maximum and minimum points, domain and range, and periodicity. For example, if the graph is a periodic wave function that has a domain from y = -3 to y = 3, it is a sine wave. If the graph has a single vertex and a strictly increasing slope, it is most likely a parabola.

Write the parent function for the type of function in the graph and superimpose the graph of this function over the original graph. In the above example, the original graph is a sine curve, so write the function p(x) = sin x and graph the curve y = sin x on the same axes as the original graph.

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Compare the positions of the two graphs to determine whether the original graph is a horizontal or vertical shift of the parent function. A function has a horizontal shift of h units if all values of the parent function (x, y) are shifted to (x + h, y) A function has a vertical shift of k if all values of the parent function at (x, y) are shifted to (x, y + k).

Adjust the graph of the parent function to match the vertical and horizontal shift in the original graph. In the above example, if the function has a vertical shift of 1 and a horizontal shift of pi, adjust the parent function p(x) = sin x to p1(x) = A sin (x - pi) + 1 (A is the value of the vertical stretch, which we have yet to determine).

Compare the orientation of the two graphs to determine whether the original graph is a reflection of the parent function along the x or y axis. The graph is a reflection along the x axis if all points (x,y) of the parent function have transformed into (x,-y). The graph is a reflection along the y axis if all points (x,y) of the parent function have transformed into (-x, y).

Adjust the function p1(x) to show a reflection along the y axis by replacing all values of x with -x. Adjust the function p1(x) to show a reflection along the x axis by changing the sign of the entire function. In the above example, if the original graph is a reflection along the y axis, change p1(x) to equal A sin (-x - pi) + 1.

Choose a point along the original graph and plug the values of x and y into the function p1(x). For example, if the sine curve passes through the point (pi/2, 4), plug in those values into the function to get 4 = A sin (-pi/2 - pi) + 1.

Solve the equation for A to find the vertical stretch of the graph. In the above example, subtract 1 from both sides to get A sin(-3 pi / 2) = 3. Replace sin(-3 pi/2)) with 1 to get the equation A = 3.