⚙️ Function Kingdom · Functions
Graphs, Tables & Equations
Move fluently between the four faces of a function: equation, table, graph and words.
In short
- Equation, table, graph and words are four views of one relationship; fluency means moving between any two.
- A constant difference in a table means the relationship is linear, and that difference is the slope.
- A graph is drawn from the y-intercept plus repeated rise-over-run steps.
- In a word problem the starting amount is b and the change per step is m, negative when the amount decreases.
One function, four faces
A function can be presented as an equation, a table, a graph or a description in words. These are not four different things — they are four views of the same relationship, and each one makes something different easy to see.
Words: "start with 3 gold and earn 2 per quest" Equation: y = 2x + 3 Table: x 0 1 2 3 y 3 5 7 9 Graph: a line through (0, 3) rising 2 for every 1 across
The equation is what you compute with. The table shows actual pairs. The graph shows the overall shape and behaviour at a glance. The words tell you what the numbers mean.
Fluency here means being able to start from any one of the four and produce the other three.
From a table to an equation
Check the differences first. If y changes by the same amount every time x increases by the same step, the relationship is linear.
x 0 1 2 3 y 7 11 15 19
y goes up 4 each time x goes up 1, so m = 4. When x = 0, y = 7, so b = 7, and the equation is y = 4x + 7.
If the table does not include x = 0, work backwards. With
x 2 3 4 y 1 4 7
the slope is 3, so y = 3x + b. Substituting (2, 1) gives 1 = 6 + b, so b = -5 and y = 3x - 5.
Always test your equation against a row you did not use.
From an equation to a table or a graph
To build a table, substitute a handful of x-values, one at a time. Choosing 0, 1, 2 and 3 makes the pattern easiest to see, and negatives are worth including once you are comfortable.
To draw a graph from y = mx + b:
1. Plot the y-intercept at (0, b). 2. From there step 1 to the right and m up (or down, if m is negative), and plot a second point. 3. Draw the line through them, and check a third point to catch slips.
Reading a value from a graph works the same way in reverse: go along to the x-value, up (or down) to the line, and across to the y-axis.
From words to an equation
Translating a description is a matter of finding two numbers.
- What is there before anything happens? That is b.
- What happens per step? That is m, negative if the amount decreases.
"A torch has 90 minutes of fuel and loses 6 minutes per chamber explored" gives m = -6 and b = 90, so y = -6x + 90.
Naming the variables in words first is worth the effort: "let x be the number of chambers and y the minutes of fuel left". It makes the answer interpretable, and it makes questions like "when does the torch run out?" easy — set y = 0 and solve, giving x = 15 chambers.
Matching representations is the same skill in disguise: to decide which equation fits a table or a graph, test one input in each candidate and discard the ones that miss.
Worked examples
Example 1
Write the equation for this table. x: 0, 1, 2, 3 and y: -2, 3, 8, 13.
- Check the differences in y: 3 - (-2) = 5, 8 - 3 = 5, 13 - 8 = 5. Constant, so linear.
- x increases by 1 each row, so the slope is 5/1 = 5.
- The row x = 0 gives y = -2, so the y-intercept is -2.
- y = 5x - 2. Check with x = 3: 15 - 2 = 13. Correct.
Example 2
A line passes through (0, 6) and (4, -2). Find its equation and the value of y when x = 5.
- Slope: (-2 - 6) / (4 - 0) = -8 / 4 = -2.
- It crosses the y-axis at 6, so b = 6 and the equation is y = -2x + 6.
- Substitute x = 5: y = -2(5) + 6.
- y = -10 + 6 = -4.
Practice problems, with solutions
Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.
Problem 1
Difficulty 1 of 5The table shows a linear relationship between x and y at x = 0, 1, 2, 3, 4. Write y as an expression in x (for example 2x - 5).
Answer: 4x + 2
- Change in y per row: 4; change in x per row: 1.
- Slope m = 4 / 1 = 4
- Use a row to find b: 2 = 4(0) + b, so b = 2.
- y = 4x + 2
Problem 2
Difficulty 3 of 5y = 6x + 8 The table of values has one entry missing. What is y when x = 0?
Answer: 8
- y = 6(0) + 8
- = 0 + 8
- y = 8
Problem 3
Difficulty 4 of 5A torch starts with 63 minutes of fuel and loses 19 minutes of fuel for every chamber Zara explores. Write the minutes of fuel left y as an expression in x, where x is the number of chambers explored.
Answer: -19x + 63
- At x = 0 (no chambers yet) the amount is 63, so b = 63.
- Each chamber changes it by -19, so m = -19.
- y = -19x + 63
Common mistakes
- Reading the input and output columns of a table the wrong way round.
- Assuming the first y-value in a table is the y-intercept even when x does not start at 0.
- Adding the constant before multiplying by the slope when filling in a table.
- Swapping m and b when matching an equation to a graph.
What you should be able to do
- Build a table of values from an equation.
- Read a slope and intercept off a graph or a table.
- Match a graph to its equation.
- Describe a function in words from any representation.
Where this fits in the curriculum
Common Core
- 8.F.A.2
Grade 8 — Compare properties of two functions represented differently — algebraically, graphically, numerically or in words.
- 8.F.B.5
Grade 8 — Describe qualitatively the relationship a graph shows, and sketch a graph from a verbal description.
- HSF-IF.B.4
High school — Interpret the key features of a graph or a table in terms of the quantities they describe.
Ontario
- MTH1W.C4.1
Grade 9 de-streamed — Compare the characteristics of graphs, tables of values and equations of linear and non-linear relations.
- MTH1W.C3.2
Grade 9 de-streamed — Move between tables of values, graphs and equations for a linear relation.
SAT
- Heart of Algebra
Moving between the equation, table and graph of a linear relationship.
- Problem Solving and Data Analysis
Reading and interpreting graphs, tables and scatterplots.