Before You Start Function notation Basic graph reading Substitution into formulas What You Will Learn Describe how A, B, C, and D change the graph of F(x)=Af(B(xC))+D. Predict a transformation before drawing the graph. Connect transformations with composition and inverse functions. Preview Activity: Move a Parent Function Start with the parent function f(x)=x^2. Instead of memorizing rules, change one number at a time and describe what you see. 1. Compare f(x)=x^2 with g(x)=2x^2. What happens to each yvalue? 2. Compare f(x)=x^2 with h(x)=(x3)^2. Which direction does the graph move? 3. Compare f(x)=x^2 with k(x)=(x+1)^2+2. List the transformations in the order they affect the graph. Hint: Choose one input, such as x=2, and compare the output before and after the change. Solution: g(x) stretches the graph vertically by a factor of 2. h(x) shifts the graph 3 units to the right. k(x) shifts left by 1, reflects across the xaxis, and shifts up by 2. The Transformation Template F(x)=Af(B(xC))+D A changes the output of the parent function, so it creates a vertical stretch or reflection. B changes the input before the parent function acts, so it creates a horizontal stretch or reflection. C moves the graph horizontally and D moves it vertically. Visual Comparison The transformed graph y=2(x1)^2+1 is the parent parabola shifted right, stretched vertically, and moved up. Why Horizontal Changes Feel Backwards In f(x3), the graph moves right because the input must be 3 larger before the parent function receives the same value. For example, f(x3) behaves like f(0) when x=3. That is why the old point at x=0 appears at x=3. Worked Example: Read a Formula Describe the transformations in g(x)=3f(2(x+4))5. Solution: The graph shifts left 4 units, compresses horizontally by a factor of 2, reflects across the xaxis, stretches vertically by 3, and shifts down 5 units. Inverse Functions f(f^{1}(x))=x \quad \text{and} \quad f^{1}(f(x))=x An inverse function undoes the action of the original function. Graphically, the inverse is reflected across the line y=x. If the reflected graph fails the vertical line test, the original function needs a domain restriction. Summary Transformations are easier when you separate input changes from output changes. Changes outside f affect yvalues; changes inside f affect xvalues. An inverse reverses a function, but it only exists when each output comes from one input. Try These 1. Describe the graph of y=4(x2)^27 from the parent function y=x^2. 2. Find the inverse of f(x)=3x8. 3. Explain why f(x)=x^2 has no inverse unless its domain is restricted.