Workshop

⚹   The y-intercept

The y-intercept is the point where a line crosses the y-axis. The y-axis is the vertical axis — where x = 0. So the y-intercept is always the point on the line with x = 0.

⚹   Calculate Without a Graph

Often you won't have a graph drawn for you. You'll need to find the intercepts from the equation alone.

Remember: the y-intercept is where x = 0, and the x-intercept is where y = 0.

Both are just substitution problems, like Chapter I. Put 0 in for the appropriate letter, then solve.

⚹   Reading Coordinates

Every point on the plane has an address: two numbers in a pair, written (x, y).

The first number is how far across (x). The second is how far up or down (y).

A glowing point sits on the plane below. Read its coordinates — count across first, then up.

⚹   Placing Points

Now reverse it. You're given a coordinate pair. Click on the plane to place a point at that address.

Your click will snap to the nearest integer grid position — aim for the correct square.

⚹   Two Points Define a Line

Place both points, then watch the line draw itself through them.

This is the key idea: any two distinct points on the plane determine exactly one straight line. You only ever need two.

⚹   Plot a Line from a Table

Here's where everything comes together. You're given a linear equation and a table of x-values.

Fill in the y-values (Chapter I skill), plot each point (Chapter III skill), then watch the line emerge.

Every point you plot will lie exactly on the line — because the equation is the line.

⚹   The One

Before any numbers, let's meet a reference line. This line has gradient 1.

It's the diagonal that goes up one square for every one square across. A perfect 45°.

Before we go further: you'll see several lines. For each, tell me whether its gradient is 1 or is not 1.

Look for the 45° climb. If it's steeper or shallower, it's not 1.

⚹   Steeper or Shallower

Gradient 1 is your anchor. Every other positive gradient is either steeper than 1 or shallower than 1.

A line with gradient 2 climbs twice as fast — it's steeper. A line with gradient ½ only climbs half as fast — it's shallower.

For each line below, pick the relationship. The faint dashed line is gradient 1 for reference.

⚹   Going Down

So far every line went up as you moved right. But lines can go down as you move right too.

A downward line has a negative gradient. The mirror of gradient 1 is gradient −1 — the 45° diagonal going down.

For each line below, pick which one it is: going up, going down, or flat.

⚹   Rise ÷ Run

Now for exactness. Every gradient is a ratio: how much the line rises per step across.

Two points on the line are marked. A staircase shows the path between them — first the run across, then the rise up (or down).

Gradient = rise ÷ run. Count the squares and calculate.

⚹   The Gradient Lives in the Equation

When a linear equation is written as y = mx + c, the gradient is already there — it's the coefficient of x.

For y = 2x + 4, the gradient is 2. For y = −3x + 1, it's −3.

Look at the equation and its line below. Tell me the gradient.

⚹   Two Points Make a Line — Table Method

A straight line is determined by any two points on it. So to graph an equation, you don't need a big table — you just need two x-values, their y-values, and a ruler.

Pick any two x-values, substitute (Chapter I), plot, draw through them. This works for any linear equation. In the next stage you'll see the two easiest x-values to pick — but this general method always works.

⚹   Plot from Intercepts — the Easy Two Points

Any two points work, but the two easiest points are the intercepts: the places where the line crosses each axis. One coordinate of each is zero, so the arithmetic is easier.

Find both intercepts for the given equation, then click the plane to plot each one. When both are placed, I'll draw the line through them.

⚹   Equation from the Line

Now the reverse. I draw a line; you read off the intercepts, then identify the equation.

Hint: the y-intercept tells you c in y = mx + c. And once you know both intercepts you can compute the gradient as rise÷run.

⚹   Equations in General Form — Substitute & Solve

Equations like 2x + y = 5 describe lines too — they just don't look like y = mx + c. This is called general form. To find y for a given x, two steps:

⚹   Rearrange to Slope-Intercept Form

Given an equation in general form, rewrite it as y = mx + c. Enter m (the coefficient of x, including sign) and c (the constant, including sign).

⚹   Intercept Method on General Form

This is the alternative technique — no rearrangement needed. Set x = 0 and solve for y — that's the y-intercept. Set y = 0 and solve for x — that's the x-intercept. Two points, one line.

Often faster than rearranging for general form, because you never have to divide by the coefficient of y. Click both intercepts on the plane and the line is drawn.

⚹   Every Linear Scenario Has Two Numbers

Real-world problems with a constant rate of change produce straight-line graphs. Each one has exactly two key numbers:

· a starting value (what you have at the beginning, when x = 0)
· a rate (how much the quantity changes per unit of x)

For each scenario below, identify those two numbers.

Sign matters: if the quantity is decreasing, the rate is negative.

⚹   Predict Using the Equation

Given a real-world equation, answer a prediction question. Some questions give you x and ask for y (substitute). Others give you y and ask for x (solve).

⚹   Read the Graph

Now in reverse. You get a graph of a real-world scenario; you answer a question by reading the line. The gradient is the rate, the y-intercept is the starting value, and any point on the line ties x and y together.

⚹   Parallel Lines — Same Gradient

Two lines are parallel when they have the same gradient. Same m, different c — the lines run alongside each other forever without meeting. Read the gradient from each equation and check whether they match.

⚹   Find Both Gradients

Given one line's gradient, enter the gradient of any line parallel to it, and the gradient of any line perpendicular to it.

You can type decimals (e.g. -0.5) or fractions (e.g. -1/2).

⚹   In Equations

Two equations, no graph. Decide the relationship. Some equations are in general form (ax + by = c) — rearrange first if needed (Chapter VIII).