MCR3U Functions · Unit 1 — Introduction to Functions
Lesson 1.3
Parent Functions
Six basic graphs are the building blocks of almost every function you will meet. Learn their equations, their shapes, and why each one looks the way it does.
Estimated time: 45 minutes
Learning goals
- recognize the six parent functions and their equations
- match an equation to its graph
- describe the shape and key features of each parent function
- explain why √x and 1/x behave differently from the others
- connect parent function shapes to real situations
Step 1 of 4
Warm Up
Evaluate a function before you graph one.
In Lesson 1.2 you evaluated functions using notation. Two quick ones, then you'll see what those outputs look like as a picture.
Step 2 of 4
Learn
Meet the six parent functions, one at a time, and see why each shape happens.
Real-world idea
Six basic shapes, like six basic recipes
Think of a pizza shop. There are only a few base recipes — plain, veggie, meat — and everything on the menu is one of those with toppings added.
Functions work the same way. There are a small number of basic graph shapes, called parent functions. Every other function you meet in this course is one of those shapes, stretched, flipped or moved.
Key term
Parent function
The simplest function of its family. Learning the six parent shapes means you can predict what a whole family of graphs looks like.
f(x) = x² is the parent of y = 3(x − 2)² + 5.
1. Linear — f(x) = x
Every output is the same as its input: 1 → 1, 2 → 2, 3 → 3. Each step right is one step up, so the points form a straight line through the origin.
You see this whenever something costs the same amount every time, like $1 per item.
- f(x) = x
2. Quadratic — f(x) = x²
Squaring makes every output zero or positive, so the graph never dips below the x-axis. It falls, turns at the origin, and rises again — a U-shaped curve called a parabola.
The area of a square patio behaves this way: as the side length grows, the area grows faster and faster.
- f(x) = x²
Worked example
Build a table of values for f(x) = x² from x = −3 to x = 3, then describe the shape.
Negative inputs
Squaring removes the sign.
(−3)² = 9, (−2)² = 4, (−1)² = 1
Zero
The smallest possible output.
0² = 0
Positive inputs
The same values mirrored.
1² = 1, 2² = 4, 3² = 9
Describe
Outputs fall to 0 then rise again, symmetric about the y-axis.
Answer: A U-shaped parabola with its lowest point (the vertex) at (0, 0).
3. Cubic — f(x) = x³
Cubing keeps the sign of the input: 2³ = 8, but (−2)³ = −8. So the graph goes down on the left and up on the right, passing through the origin in an S shape.
Volume works this way — doubling the side of a cube makes it eight times bigger.
- f(x) = x³
4. Square root — f(x) = √x
There is no real number you can square to get a negative, so negative inputs simply have no output. The graph starts at (0, 0) and only goes right.
It also rises more and more slowly: √1 = 1, √4 = 2, √9 = 3. The input jumps by 3, then 5, but the output only climbs by 1 each time.
- f(x) = √x
5. Reciprocal — f(x) = 1/x
Share a $60 pizza order among n friends and each pays 60/n. Two friends pay $30 each, four pay $15, ten pay $6. The more people share, the less each one pays — but the cost never reaches zero.
The parent version is f(x) = 1/x. At x = 0 the function is undefined, because dividing by zero has no answer, so the graph breaks into two separate branches.
- f(x) = 1/x
Undefined is not the same as zero
6. Absolute value — f(x) = |x|
Absolute value means distance from zero, so it always gives a positive answer or zero: |−7| = 7 and |7| = 7.
That folds the left half of the line upward, giving a V shape with a sharp point at the origin.
- f(x) = |x|
See them together
Tap the names below to show or hide each graph. Compare how quickly each one rises, and notice which ones exist on the left side of the y-axis.
Tap a function to show or hide it. Comparing shapes on one plane makes the families easier to tell apart.
Linear
f(x) = x
Quadratic
f(x) = x²
Cubic
f(x) = x³
Square root
f(x) = √x
Reciprocal
f(x) = 1/x
Absolute value
f(x) = |x|
Learn six shapes, and you can predict thousands of graphs.
Every function in this course is a parent function that has been stretched, flipped or moved.
Step 3 of 4
Practice
Match, evaluate, model and explain. Hints wait until you ask for them.
Build it
Name the shapes and evaluate.
Apply it
Choose the shape that models each situation.
Challenge
Use reasoning, not guessing.
Explain it
Justify, compare and correct.
Step 4 of 4
Mastery Check
Five questions, no hints. Answer them all, then see how you did.
Lesson 1.3 mastery check
5 questions · no hints during the check. Answer them all, then check your score — you can retry as many times as you like.