Function transformations and inverses (Algebra II) · lesson

Shifting left, right, up, and down

💡 A change inside the parentheses moves the graph sideways the opposite way; a change outside moves it up or down.

f(x) + k moves the graph up k. f(x) − k moves it down k.

f(x − h) moves the graph right h. f(x + h) moves it left h. The sign feels backward because x has to be h bigger to give the same output.

The vertex of y = x² is (0, 0). It moves with the graph, so (x + 5)² − 1 has its vertex at (−5, −1).

🧠 Let's work it out

Shift f(x) = x² 5 units left and 1 unit down.

What do you think comes next?

💡 A minus sign outside flips the graph over the x-axis, and a number outside stretches it up and down.

−f(x) changes the sign of every output, so the graph flips over the x-axis. f(−x) changes the sign of every input, so it flips over the y-axis.

3f(x) multiplies every output by 3, a vertical stretch. A number between 0 and 1 squashes the graph instead.

With several changes, work from the inside out: first the shift inside, then the stretch and flip outside.

🧠 Let's work it out

f(x) = x² and g(x) = −3f(x − 2). Write the rule for g(x).

What do you think comes next?

💡 To find f(something), replace every x in the rule with that something, then simplify.

If f(x) = x² + 3, then f(3x) = (3x)² + 3 = 9x² + 3. Use parentheses so the whole expression gets squared.

For a rule like f(x + 1) − 4, put (x + 1) in for every x, then subtract 4. Expand and combine like terms.

🧠 Let's work it out

f(x) = x² − 2x and g(x) = f(x + 1) − 4. Write the rule for g(x).

What do you think comes next?

💡 To find an inverse, swap x and y, then solve for y.

An inverse function undoes the original. If f adds 10 then divides by 5, the inverse multiplies by 5 then subtracts 10.

The steps: write y = f(x). Swap x and y. Solve the new equation for y. That y is f⁻¹(x).

🧠 Let's work it out

Find the inverse of f(x) = 6/(x + 2).

What do you think comes next?

💡 If (a, b) is on the graph of f, then (b, a) is on the graph of f⁻¹.

f takes a to b, so f⁻¹ takes b back to a. Every point on f has its coordinates swapped on f⁻¹.

On a graph, this is a reflection across the line y = x.

🧠 Let's work it out

f(x) = 3^x, and (2, 9) is on its graph. Find a point on the graph of f⁻¹.

What do you think comes next?