Linear Functions: Graphs and Properties

Try this

Drag only $b$ and leave $k$ alone. Does the tilt of the line change? What kind of motion is it?
Drag only $k$ and leave $b$ alone. Which point does the line turn around? How are that point's coordinates related to $b$ ?
Look at the small dashed triangle next to the line: go $1$ square to the right, and how far does the line go up (or down)? Set $k$ to $2$ , $- 1$ and $0.5$ and look each time.
Make $k$ positive and $b$ negative. Which quadrant does the line miss? Try the other three sign combinations and note the results.

What you just saw is the whole personality of the linear function $y = k x + b$ (with $k \neq = 0$ ):

y = k x + b

$k$ is the slope: each time $x$ increases by $1$ , $y$ changes by $k$ . For $k > 0$ the line goes up; for $k < 0$ it goes down.
$b$ is the intercept: the line meets the $y$ -axis at $(0, b)$ . Changing $b$ slides the line up or down.

Predict first

Without drawing, just from the signs: which quadrant does the line $y = - 2 x + 3$ not pass through?

Quadrant IQuadrant IIQuadrant IIIQuadrant IV

Choose an answer first; then you can see the result.

Why: reading the quadrants from k and b

No rhyme to memorize. A line has only two “directions” and three “starting heights”; just go one step at a time.

Example: which quadrants does y = 2x − 3 pass through?

Step 1 / 4

Start with the simplest case: b = 0
$y = 2 x$ goes through the origin. $k = 2 > 0$ , so it heads up to the right and passes only through quadrants I and III.

$y = 2 x$
Add b: the whole line slides
Find where it crosses the x-axis
Read off the quadrants

The pattern fits in two sentences: the sign of $k$ decides between quadrants I and III or quadrants II and IV; the sign of $b$ decides which extra quadrant it clips: one of the top two or one of the bottom two.

Two points fix a line

$y = k x + b$ has two unknown numbers, $k$ and $b$ , so two points on the line are enough. Writing down the form first and then using the conditions to pin down the coefficients is called the method of undetermined coefficients.

Example: find the line through (−1, 4) and (2, −2)

Step 1 / 4

Write down the form
Let the line be $y = k x + b$ . The picture starts with any old line: both circles show ✗.

$y = k x + b$
Find k first: how do you get from one point to the other?
Then find b: substitute one point
Check with the other point

One problem, several solutions

A line passes through $(1, 3)$ and $(4, 9)$ . Find its equation.

Set up a system

Let $y = k x + b$ and substitute both points:

{k + b = 3 4 k + b = 9

Subtracting gives $3 k = 6$ , so $k = 2$ ; substituting back gives $b = 1$ . So $y = 2 x + 1$ .

CostSafe and works for any two points; you have to solve a system of two linear equations.

Slope first

From $(1, 3)$ to $(4, 9)$ : $3$ to the right and $6$ up, so $k = \frac{9 - 3}{4 - 1} = 2$ .

Step back $1$ square from $(1, 3)$ to the $y$ -axis and the height drops by $2$ , so $b = 3 - 2 = 1$ . So $y = 2 x + 1$ .

CostFast, and you can see it in the picture; keep the order “difference in y ÷ difference in x” the right way round.

Where two lines meet

The intersection point lies on both lines, so its coordinates satisfy both equations at once: the intersection point is the solution of the system.

The dashed line is

y = - x + 4

. Drag the solid line: the intersection readout is the solution of the system. What happens if you set

k

- 1

When two lines have equal $k$ but different $b$ , they are parallel and never meet: the matching system has no solution.

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: The line $y = k x + b$ meets the $x$ -axis exactly once

As long as $k \neq = 0$ , the equation $k x + b = 0$ always has exactly one solution, $x = - \frac{b}{k}$ . Drag all you like: “Meets x-axis” always shows a value.

Practice

Exercise 1Basic

What is the $x$ -coordinate of the point where the line $y = 3 x - 6$ meets the $x$ -axis?

Exercise 2Basic

A line passes through $(- 2, 7)$ and $(3, - 3)$ . Write the expression that comes after $y =$ .

Exercise 3Advanced

The line $y = 2 x + 4$ and the two coordinate axes enclose a triangle. What is its area?

Exercise 4Advanced

$y = (m - 3) x^{∣ m ∣ - 2} + m + 1$ is a linear function. Find $m$ .

Exercise 5Challenge

Whatever value $k$ takes, the line $y = k x + 2 k + 1$ always passes through the same point. What are that point's $x$ - and $y$ -coordinates? (Separate the two numbers with a comma.)

Try this

Why: reading the quadrants from k and b

Example: which quadrants does y = 2x − 3 pass through?

Two points fix a line

Example: find the line through (−1, 4) and (2, −2)

Set up a system

Slope first

Where two lines meet

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Checking the exponent and forgetting k ≠ 0

Treating an intercept as a distance