Graph Gallery

Graph Gallery

Two skills carry nearly all the marks on graphs, and only one of them is about drawing. The first is being able to look at y = mx + c and say instantly how steep the line is and where it crosses the axis, and to work backwards from a pair of points to the equation itself. The second is recognising a curve's shape from its equation without plotting a single point. The trap in between is that most exam equations do not arrive in the tidy form the rule expects.

The words and letters

Everything in this module is one of these five things:

Plot the coordinate

Plot the point (−4, 3): 4 units LEFT of the origin and 3 units UP. Tap it on the grid.

Gradient from two points

A line passes through the points (1, 3) and (4, 12). What is its gradient?

Watch the order

What is the gradient of the line y = 5 − 2x?

• −2
• 2
• 5
• −5

Rearrange before you read

The rule "m is the number in front of x" only works once the equation actually looks like y = mx + c. Take 2y = 4x + 6:

Rearrange, then read

What is the gradient of the line 3y = 12x − 9?

Which is parallel?

Which of these lines is parallel to 2y = 6x + 4?

• y = 3x − 7
• 3y = 6x + 4
• y = 2x + 3
• y = −3x + 2

From two points to an equation

A line passes through (2, 1) and (5, 10). Find its equation. Step 1, the gradient. rise ÷ run = (10 − 1) ÷ (5 − 2) = 9 ÷ 3 = 3. So far: y = 3x + c. Step 2, the intercept. Both points are on the line, so both must satisfy the equation. Put (2, 1) in: 1 = 3 × 2 + c, so 1 = 6 + c, so c = −5. The equation is y = 3x − 5. Step 3, check it. Use the OTHER point, the one you did not substitute: 3 × 5 − 5 = 10, and the point was (5, 10). It fits, so the answer is right. And what the gradient means. On a graph of cost against number of items, a gradient of 3 means each extra item costs 3. The gradient is always a rate: how much y changes for every one that x does.

Find the equation

A line has a gradient of −2 and passes through the point (3, 1). What is its equation?

• y = −2x + 7
• y = −2x − 5
• y = −2x + 1
• y = −2x + 3

Read the shape off the powers

You do not need to plot anything to know what a graph looks like. Find the highest power of x and the shape follows: • Highest power x¹ (or no x term at all beyond a single x): a straight line. • Highest power x²: a parabola, the U shape. If the x² term is negative, the U is upside down. • Highest power x³: a cubic, the S-shaped curve. • x on the bottom of a fraction, as in 1/x: a reciprocal, in two separate branches that never touch the axes. Everything else in the equation only shifts, stretches or flips the shape. y = x², y = x² − 4 and y = 3x² + 7x − 1 are all parabolas, because the highest power is x² in every one of them. Adding, subtracting and multiplying by constants cannot change what kind of curve you have.

Find it in the gallery

Four graphs. Tap the one drawn from an equation whose highest power of x is two.

What shape is each one?

Draw it from the equation

Put the steps for drawing a line from y = mx + c into order.

• Rearrange the equation until y is on its own, if it is not already
• Mark c on the y-axis: that is one point on the line, free of charge
• From there, count 1 across and m up or down, according to the sign
• Mark that second point
• Draw a straight line through both, extending it to the edges of the grid

The equation of a line

A straight line has the equation y = mx + c, but only once y is on its _____. Before that, the number in front of x is not the gradient. The letter m is the gradient, found as rise ÷ run, and it is also a _____: how much y changes for every one that x does. The letter c is the _____. Two lines are parallel when their gradients are equal, sign included. And to name a curve's shape, look for the highest _____ of x.