Unit 1: Function Families & Transformations

Unit 1 Resources & Lessons

VA SOL Standard: AFDA.AF.1. This unit focuses on understanding function transformations conceptually and visualizing them using online tools like the Desmos Graphing Calculator.

Open Desmos Calculator

Daily Lesson Slides & Worksheets

Day 1: Parent Functions

View Slides Print Worksheet

Day 2: Translations (Shifts)

View Slides Print Worksheet

Day 3: Reflections

View Slides Print Worksheet

Day 4: Dilations & Combining

View Slides Print Worksheet

Extra Practice

Unit 1 Practice Worksheet Unit 1 Study Guide

Day 1: Parent Functions

Objective: To identify the graphs and equations for linear, quadratic, and exponential parent functions and view them in Desmos.

Notes

A parent function is the simplest, most basic version of a function in a family. Think of it as the starting point before we make any changes.

1. Linear Parent Function: $f(x) = x$. This is a straight line that passes through the origin at a 45-degree angle.

2. Quadratic Parent Function: $f(x) = x^2$. This is a U-shaped curve called a parabola, with its vertex at the origin.

3. Exponential Parent Function: $f(x) = a^x$ (we often use $f(x) = 2^x$ as the basic example). This graph starts flat and then grows very quickly.

Desmos Exploration: Graphing Parent Functions

Let's see what these look like!

  1. Go to desmos.com/calculator.
  2. In the first expression box, type y = x. You'll see the straight line of the linear parent function.
  3. In the second box, type y = x^2. You can get the ^ symbol by pressing Shift+6. This is the quadratic parent function.
  4. In the third box, type y = 2^x. This is the exponential parent function. Click the colored circles next to each equation to turn them on and off to compare.

You Do: Practice Problems

Day 2: Translations (Shifts)

Objective: To describe and apply vertical and horizontal translations to parent functions using Desmos sliders.

Notes

A translation is a slide or shift of the graph. It moves the entire graph without changing its shape.

Vertical Shift (Up/Down): Caused by adding or subtracting a number OUTSIDE the function.
$f(x) + k$ moves the graph UP $k$ units.
$f(x) - k$ moves the graph DOWN $k$ units.

Horizontal Shift (Left/Right): Caused by adding or subtracting a number INSIDE the parentheses with the $x$.
$f(x - h)$ moves the graph RIGHT $h$ units (opposite of the sign!).
$f(x + h)$ moves the graph LEFT $h$ units (opposite of the sign!).

Desmos Exploration: Using Sliders for Translations

This is where Desmos gets really powerful!

  1. Clear any old equations in Desmos.
  2. Type y = (x-h)^2 + k.
  3. Desmos will show buttons under the equation to "add slider" for h and k. Click both.
  4. Now, move the slider for k. Notice how the graph slides up and down.
  5. Move the slider for h. Notice how the graph slides left and right, and how the direction is the *opposite* of the sign of h.

You Do: Practice Problems

Day 3: Reflections

Objective: To describe and apply reflections across the x-axis and y-axis using Desmos.

Notes

A reflection is a flip of the graph over a line.

Reflection over the x-axis: Caused by a negative sign OUTSIDE the function.
Equation: $-f(x)$. This flips the graph upside down.

Reflection over the y-axis: Caused by a negative sign INSIDE the parentheses with the $x$.
Equation: $f(-x)$. This flips the graph sideways.

Desmos Exploration: Visualizing Reflections
  1. In Desmos, graph the parent function y = (x-2)^2. (We shifted it so the flip is easier to see).
  2. In the next line, type y = -(x-2)^2. Notice how the negative OUTSIDE the function flips the parabola upside down, over the x-axis.
  3. Now, in the third line, type y = (-x-2)^2. Notice how the negative INSIDE the parentheses flips the graph sideways, over the y-axis.

You Do: Practice Problems

Day 4: Dilations & Combining Transformations

Objective: To describe dilations and combine multiple transformations using Desmos.

Notes - Part 1: Dilations

A dilation is a stretch or a compression (squish) of the graph.

Vertical Dilation: Caused by multiplying a number OUTSIDE the function: $a \cdot f(x)$.
If $|a| > 1$, it's a vertical stretch (graph gets skinnier).
If $0 < |a| < 1$, it's a vertical compression (graph gets wider).

Desmos Exploration: Sliders for Dilations
  1. In Desmos, type y = a*x^2.
  2. Click to "add slider" for a.
  3. Move the slider for a. Notice how when a is bigger than 1, the graph gets skinnier (a vertical stretch).
  4. When a is between 0 and 1, the graph gets wider (a vertical compression).

Notes - Part 2: Combining Transformations

When you have multiple transformations, describe them in order:
1. Horizontal Shift (left/right)
2. Reflection / Dilation (flip / stretch / shrink)
3. Vertical Shift (up/down)

Examples

I Do: Describe All Transformations

Describe the transformations for $f(x) = -2(x-3)^2 + 5$.

  1. The parent function is quadratic: $x^2$.
  2. $(x-3)$: Horizontal shift right 3.
  3. Negative sign: Reflection over the x-axis.
  4. Number 2: Vertical stretch by a factor of 2.
  5. +5: Vertical shift up 5.
  6. Check it! Graph y = x^2 and y = -2(x-3)^2 + 5 in Desmos to confirm.
You Do: Practice Problems

Ready to test your knowledge?

You have completed the notes for Unit 1. Click the button below to take the quiz.

Take the Unit 1 Quiz