SumNinja

Read a linear graph from its start and rate

About 3 min

A straight-line rule has a starting value and a fixed change per step. Those two clues help you pick its graph.

A bike rental costs 4 coins at the start and 2 more for each hour. You want a graph that shows the full cost. Let x mean hours and y mean coins. The rule is y = 2x + 4. Each extra hour adds the same amount.

Build the cost line from its meaning

Match y = 2x + 4 to a graph.

  1. Start before any hours pass

    x=0⇒y=2(0)+4=4x=0\quad\Rightarrow\quad y=2(0)+4=4

    The start charge is four coins. The graph crosses the upright y-axis at (0, 4).

  2. Pay for one hour, then another

    x=1⇒y=6;x=2⇒y=8x=1\Rightarrow y=6;\quad x=2\Rightarrow y=8

    The cost rises by two each time x rises by one. The points (0, 4), (1, 6) and (2, 8) sit on one straight line.

  3. Use both clues together

    start=4change per hour=2\begin{gathered}\text{start}=4\\\text{change per hour}=2\end{gathered}

    A line that starts at two or rises by four does not fit. The start and the change must both match.

The start charge and hourly cost make the line

The bottom axis gives hours. The side gives coins. At zero hours the cost is four; each extra hour adds two.

Names for the two clues

The start at x = 0 is the y-intercept. The change in y for one step in x is the slope. In y = 2x + 4, the slope is 2 and the start is 4.

A function is a rule with one output for each input. Here each rental time gives one cost. The story uses zero or later hours; the mathematical line also extends the other way.

A negative slope means y falls as x rises. For y = −3x + 2, start at two and fall three per step. This could model a tank losing water rather than a bill growing.

Read numbers rather than apparent steepness

Changing the scale on an axis can make the same line look steeper. Check the axis labels and calculate a point. A familiar-looking slant is not enough.

Try it yourself

Work out an answer and explain your reasoning. Then check the explanation.

Read another rule

y=−3x+2y=-3x+2

What start and one-step change does this rule need?

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(0,2),(1,−1)(0,2),\quad(1,-1)

At x = 0, y is 2. At x = 1, y is −1, so it fell by 3.

Assign each quantity its role

y=5x+3  ?y=5x+3\;?

Write one rule for the tank. Decide which number is the start and which is the change per minute.

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y=3x+5y=3x+5

Five is the starting amount. Three is the amount added each minute.

Choose the graph, not just a number

y=x+3y=x+3

Both graphs pass through (1, 4). Which also fits the rest of the rule?

Graph A

At zero, the value is one. Each step adds three.

Graph B

At zero, the value is three. Each step adds one.

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Graph B\text{Graph B}

At x = 0 the rule gives three. Graph B starts at three and rises by one. Graph A starts at one and rises by three.

What does f(x) mean?

You may see f(x) = 2x + 4 instead of y = 2x + 4. Read f(x) as “the result when the input is x.” It does not mean f times x. For example, f(3) = 10.

Try matching a line

Use the start and slope to reject a wrong graph, then check a point. The game changes the rule and asks you to choose its graph.

2-minute practice · keep going after mistakes · opens in a new tab

Try linear graph questions