← Algebra II

Unit 9: Transformations, Composition, and Inverses of Functions

Every parent function's family tree, stretched, shifted, reflected, chained together, and run in reverse.

8 topics · 31 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Parent Functions and Their Graphs

  1. How it worksHow do the shapes of linear, quadratic, cubic, absolute value, and square root parent functions differ?
  2. Worked exampleSketch the parent functions f(x) = x2 and f(x) = the square root of x and compare their domains.
  3. Common mistakeWhy isn't every U-shaped graph automatically the parent function y = x2?
Your turn1 of 2
Compare the domains of f(x) = x2 and g(x) = the square root of x. Which statement is correct?

Vertical and Horizontal Shifts

  1. How it worksHow does replacing x with (x - h) inside a function shift the whole graph horizontally by h?
  2. Worked exampleGraph g(x) = (x - 3)2 + 2 as a shift of the parent function f(x) = x2.
  3. Common mistakeWhy does f(x - 3) shift a graph to the right instead of to the left?
  4. Going furtherWhat happens when you shift a function both vertically and horizontally at the same time?
Your turn1 of 2
The function f(x) = x2 is shifted right by 7 and up by 6. Write g(x).

Vertical and Horizontal Stretches and Compressions

  1. How it worksHow does the number a in a times f(x) scale every y-value on the graph by that same factor?
  2. Worked exampleGraph g(x) = 3 times the square root of x as a vertical stretch of f(x) = the square root of x.
  3. Common mistakeWhy does f(2x) compress a graph horizontally instead of stretching it?
  4. Ties togetherHow does stretching this square root graph use the same a-value rule that stretched an exponential function's graph?
Your turn1 of 2
The function f(x) = square root of x is vertically stretched by a factor of 7. Write g(x).

Reflecting Functions Across the Axes

  1. How it worksHow does negating f(x) reflect a graph over the x-axis, while negating x inside f reflects it over the y-axis?
  2. Worked exampleGraph g(x) = -(x + 1)2 as a reflection and shift of the parent function f(x) = x2.
  3. Common mistakeWhy doesn't -f(x) and f(-x) produce the same reflected graph?
  4. Going furtherWhat happens to a graph when you reflect it over both the x-axis and the y-axis at the same time?
Your turn1 of 2
The function f(x) = x2 is shifted left by 7, reflected across the x-axis, and shifted up by 6. Write g(x).

Combining Multiple Transformations

  1. How it worksWalk me through graphing g(x) = -2 times the square root of (x - 1) plus 3 as a chain of transformations.
  2. Worked exampleGraph g(x) = -2(x + 4)2 - 1 by transforming the parent function f(x) = x2 step by step.
  3. Common mistakeWhy doesn't performing the shift before the stretch give you the same graph as doing the stretch first?
Your turn1 of 2
The function f(x) = square root of x is shifted left by 6, stretched by a factor of 7, reflected across the x-axis, and shifted up by 8. Write g(x).

Composing Functions

  1. How it worksHow does f of g of x mean plugging the entire function g(x) in for every x in f?
  2. Worked exampleFind f(g(x)) and g(f(x)) if f(x) = 2x + 1 and g(x) = x2 - 3.
  3. Common mistakeWhy doesn't f(g(x)) usually equal g(f(x))?
  4. Going furtherWhat has to be true about the domains of f and g for f(g(x)) to even be defined at a given x?
Your turn1 of 2
Find f(g(x)) if f(x) = 7x + 6 and g(x) = x2 - 8, as a function of x.

Finding and Verifying Inverse Functions

  1. How it worksHow do you find a function's inverse by swapping x and y and solving for the new y?
  2. Worked exampleFind the inverse of f(x) = (2x - 5) over 3 and verify it by composing the two functions.
  3. Common mistakeWhy isn't a function's inverse the same thing as its reciprocal, 1 over f(x)?
  4. Going furtherWhat has to be true about a function for it to actually have an inverse that is also a function?
Your turn1 of 2
Find the inverse of f(x) = (7x - 6) over 9.

Graphing a Function and Its Inverse

  1. How it worksHow does swapping every x and y coordinate on a graph produce the graph of the inverse function?
  2. Worked exampleGraph f(x) = x3 and its inverse on the same axes, using the line y = x as a guide.
  3. Common mistakeWhy does a function like f(x) = x2 need its domain restricted before it can have a real inverse function?
  4. Going furtherWhat happens to the graph of an inverse relation when the original function isn't one-to-one over its whole domain?
  5. Ties togetherHow does restricting a function's domain so it has an inverse connect to the vertical line test that confirmed it was a function?
Your turn1 of 2
If the point (7, 6) lies on the graph of f, find the y-coordinate of the corresponding point on the graph of f−1.