A. Quadratic Functions · Quadratic FunctionsRelations and Functions: What Makes a Function?
A2.1About 65 minutes
Learning goals- I can explain the difference between a relation and a function.
- I can decide whether a relation is a function from a table, a mapping diagram, a graph, a function machine or an equation.
- I can use the vertical line test.
- I can explain why non-vertical lines and quadratics y = ax² + bx + c are functions.
Words to know
relation
A set of ordered pairs (x, y) that connects one quantity to another.
function
A relation in which each input (x-value) has exactly one output (y-value).
independent variable
The input variable, usually x. You choose its value.
dependent variable
The output variable, usually y. Its value depends on the input.
mapping diagram
A picture with inputs in one oval, outputs in another, and arrows joining each input to its output(s).
vertical line test
A graph is a function if no vertical line crosses it more than once.
function machine
A model of a function: an input goes in, a rule is applied, and one output comes out.
One button, one snack
At a vending machine you press B4 and a granola bar drops out. Press B4 again tomorrow and you expect the same granola bar, not a granola bar or a juice box. Each button gives exactly one result. Two buttons can give the same snack (B4 and B5 might both be granola bars), and that is fine.
A machine like this, where every input gives exactly one output, is what mathematicians call a function. Functions are the main idea of Grade 11 math. In this unit you will look at quadratic relations (the parabolas you met in Grade 10) through the lens of functions.
Remember from Grades 9 and 10
In Grade 9 you compared linear and non-linear relations using tables, graphs and equations, and you used first differences. In Grade 10 you studied quadratic relations y = ax² + bx + c, their parabolas, second differences and vertex form y = a(x − h)² + k. This unit gives those relations a new name, new notation and new tools. The companion unit Quadratic Expressions and Equations covers expanding, factoring and solving quadratic equations.
Relations
A relation is any set of ordered pairs (x, y). It connects one quantity to another.
- The first coordinate, x, is the input. It is also called the independent variable.
- The second coordinate, y, is the output. It is also called the dependent variable, because its value depends on x.
A relation can be shown in many ways: a list of ordered pairs, a table of values, a mapping diagram, a graph, a function machine or an equation.
Functions
A function is a special relation: each input has exactly one output.
- Two different inputs may share an output. That is allowed.
- One input may not have two different outputs. Then the relation is not a function.
In a list or table, look for a repeated x-value. If the same x appears with two different y-values, the relation is not a function.
Worked exampleExample 1: Tables and lists
Is each relation a function? Explain.
(a) {(1, 3), (2, 5), (3, 7), (4, 9)}
(b) {(2, 1), (3, 4), (2, −1), (5, 0)}
(c) {(0, 5), (1, 5), (2, 5), (3, 5)}
- Look at the inputs (the x-values) in each list.
- (a) The inputs 1, 2, 3, 4 are all different, so each input has one output. Function.
- (b) The input 2 appears twice, with outputs 1 and −1. One input has two outputs, so it is not a function.
- (c) Every input gives the output 5. Different inputs share one output, which is allowed. Function.
(a) function, (b) not a function, (c) function.
Graphs: the vertical line test
On a graph, a vertical line is the set of all points with one x-value. If a vertical line touches the graph at two or more points, that x-value has two or more outputs.
Vertical line test: a graph is a function if no vertical line crosses it more than once.
Imagine sliding a ruler, held straight up and down, across the whole graph from left to right. If it ever touches the graph twice at the same moment, the relation is not a function.
Worked exampleExample 2: Which graphs are functions?
Use the vertical line test on each graph.
(a) The parabola y = −2x² + 3
(b) The horizontal line y = 4
(c) The vertical line x = 4
(d) A circle with centre (0, 0) and radius 5
- (a) Every parabola that opens up or down is crossed at most once by a vertical line. Function.
- (b) A vertical line crosses y = 4 exactly once. Function (every input has the output 4).
- (c) The vertical line x = 4 is itself a vertical line. It lies on top of the graph, touching infinitely many points. Not a function.
- (d) The line x = 3 crosses the circle at (3, 4) and (3, −4). Not a function.
(a) and (b) are functions; (c) and (d) are not.
Equations
With an equation, ask: if I choose an x, can I get more than one y? It often helps to solve for y.
- y = 3x − 2: for each x there is one value of 3x − 2. Function. Every linear relation is a function except a vertical line x = k.
- y = x² − 4x: each x gives one value. Function. Every quadratic y = ax² + bx + c is a function.
- x = y²: solving gives y = √x or y = −√x. For x = 9, y = 3 or −3. Not a function.
- x² + y² = 25 (a circle): y = ±√(25 − x²). For x = 0, y = 5 or −5. Not a function.
A warning sign is a y² term, or anything that forces you to write ± when you solve for y.
Worked exampleExample 3: Functions from equations
Decide whether each equation defines y as a function of x.
(a) y = 5 − x²
(b) y² = x + 1
(c) 2x + 3y = 12
- (a) Choose any x. Squaring, then subtracting from 5, gives one number. Function. (It is a parabola opening down.)
- (b) Try x = 3: y² = 4, so y = 2 or y = −2. One input, two outputs. Not a function.
- (c) Solve for y: 3y = 12 − 2x, so y = 4 − (2/3)x. This is a non-vertical line. Function.
(a) function, (b) not a function, (c) function.
A function machine takes an input, applies a rule and gives one output. With the rule y = x² − 4, the input 3 always gives 9 − 4 = 5.Why functions matter
When a relation is a function, every input leads to one predictable output. That is exactly what we need for modelling: the height of a ball at 2 seconds, the cost of 50 T-shirts, the temperature at noon. A height-time graph that gave two heights at the same moment would make no physical sense.
In the next lesson, Function Notation, you will learn a short way to write "the output of this function when the input is 2".
Just for funWhy did the circle never get invited to the Function Club?
It kept giving two answers to the same question. The parabola, on the other hand, was always very straightforward (well, straight up and down).
Hands-on activityMapping the people you know
You need: paper and a pencil, and a list of 6 to 10 people you know (family, friends, teammates) with their birth months. Ask permission before you write down anyone's birthday.
- Draw a mapping diagram. Put the names in a left oval and the months in a right oval. Draw an arrow from each person to their birth month.
- Is "person → birth month" a function? (Each person has exactly one birth month, so it is, even if two people share a month.)
- Now reverse it: draw arrows from months to people. Is "month → person" a function? (It fails as soon as one month has two people.)
- Make up one more home relation and test it, for example "room → number of windows" or "day of the week → what you ate for breakfast" over two weeks.
What to look for: "person → birth month" is a function. "Month → person" is a function only if no month is shared by two people. Breakfast over two weeks is usually not a function, because Monday appears twice with possibly different breakfasts.
Your turn
1Is {(−1, 2), (0, 2), (1, 2), (2, 2)} a function? Explain.
Yes. The inputs −1, 0, 1, 2 are all different. They share the output 2, which is allowed.
2Is {(3, 1), (4, 2), (3, 5)} a function? Explain.
No. The input 3 has two outputs, 1 and 5.
3A mapping diagram has arrows 1 → 6, 2 → 7, 3 → 6 and 4 → 8. Is it a function?
Yes. Each input has exactly one arrow. Two inputs (1 and 3) share the output 6, which is allowed.
4Does the graph of y = (x − 3)² + 1 pass the vertical line test? How do you know without drawing it?
Yes. It is a quadratic of the form y = a(x − h)² + k, a parabola that opens up, and each x gives one value of y.
5Decide whether x = 7 is a function. Explain.
No. It is a vertical line. The input 7 is paired with every y-value.
6Is y² = 4x a function? Test x = 1.
For x = 1, y² = 4, so y = 2 or y = −2. One input has two outputs, so it is not a function.
7Name a real-life relation that is a function and one that is not.
Many answers. Function: "student → student number" (each student has one number). Not a function: "age → height" for a class (two 16-year-olds can have different heights).
More practice: warm-up
1What is the input of the ordered pair (5, −2)? What is the output?
2Is {(1, 1), (2, 4), (3, 9)} a function?
3Is {(0, 0), (0, 1)} a function?
No. The input 0 has two outputs.
4Does a horizontal line pass the vertical line test?
Yes. Every vertical line crosses it exactly once.
5Is y = 2x + 5 a function?
Yes. It is a non-vertical line: each x gives one y.
More practice: on your own
1A relation is given by the table x: 2, 4, 6, 4 and y: 1, 3, 5, 3. Is it a function? Careful!
Yes. The input 4 appears twice, but both times with the same output 3. That is just the same pair listed twice, so each input still has exactly one output.
2Decide whether x² + y = 9 is a function.
Solve for y: y = 9 − x². Each x gives one y, so it is a function (a parabola opening down).
3Decide whether x² + y² = 9 is a function.
Solve: y = ±√(9 − x²). At x = 0, y = 3 or −3. Not a function (it is a circle).
4A parabola opens to the right with vertex (0, 0). Is it a function? Explain with a point.
No. For example, x = y² opens right; the vertical line x = 1 meets it at (1, 1) and (1, −1).
5A table gives the height of a maple seedling every month: month 1 → 4 cm, month 2 → 7 cm, month 3 → 9 cm, month 4 → 9 cm. Is height a function of month? Explain.
Yes. Each month has one height. Months 3 and 4 share 9 cm (the seedling stopped growing), which is allowed.
Challenge
1Explain why every quadratic y = ax² + bx + c is a function, but not every parabola is.
For y = ax² + bx + c, any x gives one value of ax² + bx + c, so each input has one output. A parabola can also open sideways, like x = y², and then a vertical line crosses it twice. So "parabola" alone does not guarantee a function; the equation must give y in terms of x.
2Write a set of four ordered pairs that is a function, and whose reverse (swap every x and y) is not a function.
Example: {(−2, 4), (−1, 1), (1, 1), (2, 4)}. It is a function (no repeated inputs). Swapped: {(4, −2), (1, −1), (1, 1), (4, 2)}, where the input 1 (and 4) has two outputs, so it is not a function.
3Is the relation "x → a number whose square is x" a function for x ≥ 0? How could you change it to make it a function?
No: for x = 4 the outputs could be 2 or −2 (for x > 0 there are always two). Restricting to the positive (non-negative) square root, y = √x, makes it a function, because each x ≥ 0 has exactly one non-negative square root.
Quiz
1. Which statement defines a function?
2. Which relation is not a function?
3. The vertical line test checks whether…
4. Which equation defines y as a function of x?
5. A circle with centre (0, 0) passes through (0, 3). Why is it not a function?
6. For which relation is it fine that two inputs have the same output?
Big idea: expectation A2.1 asks students to explain the meaning of function and to tell functions from relations that are not functions, using tables of values, mapping diagrams, graphs, function machines and equations.
How to help: turn everyday pairings into quick questions: "Is person → phone number a function? What about phone number → person for a shared family phone?" Ask your teen to give a reason every time ("one input, one output").
Common difficulties:
- Thinking repeated outputs break the rule. Only a repeated input with different outputs does.
- Forgetting that a vertical line x = k is a relation but not a function.
- Missing the ± when solving for y in equations such as y² = x.
- Listing the same ordered pair twice and calling it "not a function": (4, 3) twice is still one output for 4.