GradeGen.AI

Recognising and Interpreting Linear Graphs

Every time you take a taxi and pay a starting fare plus a fixed cost per mile, you are experiencing a linear function in real life.

A linear function creates a straight line and is written in the general form $y = mx + c$ .
The gradient ( $m$ ) represents the steepness of the line and the constant rate of change (e.g., cost per mile).
The y-intercept ( $c$ ) is the coordinate where the line crosses the -axis, often representing a fixed starting value when .

Direction and Special Lines

A positive gradient ( $m > 0$ ) goes "uphill" from bottom-left to top-right, while a negative gradient ( $m < 0$ ) goes "downhill" from top-left to bottom-right.
A larger absolute value of $m$ creates a steeper line (e.g., $-5$ is steeper than $-4$ ).
Horizontal lines are written as $y = k$ (where gradient is zero), and vertical lines are written as $x = k$ (where gradient is undefined).
Parallel lines have identical gradients ( $m_1 = m_2$ ), while perpendicular lines (Higher Tier) have gradients that are negative reciprocals ( $m_2 = -1/m_1$ ).

Worked Example: Calculating Gradient

Calculate the gradient of the line passing through $A(7, -1)$ and $B(4, 5)$ .

Step 1: State the formula for the gradient using two points.

m = \frac{y_2 - y_1}{x_2 - x_1}

Step 2: Substitute the coordinates, using brackets for negative numbers to avoid sign errors. $m = \frac{5 - (-1)}{4 - 7}$

Step 3: Calculate the final value. $m = \frac{6}{-3} = -2$

Sketching Linear Graphs

To plot a graph, you must accurately calculate points using a table of values (using at least 3 points to verify the straight line).
To sketch a graph, you only need to draw an approximate diagram showing the correct direction and labelling key intersections with the axes.

Worked Example: Sketching a Linear Graph

Sketch the graph of $y = -2x + 4$ .

Step 1: Identify and label the $y$ -intercept. The equation is in the form $y = mx + c$ , so $c = 4$ . Label the point $(0, 4)$ on the -axis.

Step 2: Find the $x$ -intercept by setting $y = 0$ . $0 = -2x + 4$ $2x = 4$ , so label the point on the -axis.

Step 3: Draw a straight line through these points. Since $m = -2$ , the line correctly slopes downhill from top-left to bottom-right.

Recognising and Interpreting Quadratic Graphs

You can easily predict a constant speed with a straight line, but understanding accelerating objects requires a curve.

A quadratic function ( $y = ax^2 + bx + c$ ) produces a smooth, symmetrical curve called a parabola.
The shape is determined by the quadratic coefficient ( $a$ ).
If $a > 0$ , the graph is with a minimum at its lowest value.

Key Features of a Parabola

Roots: These are the $x$ -intercepts where the graph crosses the $x$ -axis (found by setting $y = 0$ ). A parabola can have 2, 1, or 0 roots.
y-intercept: This occurs at $(0, c)$ where the graph crosses the $y$ -axis.

Sketching Quadratic Graphs

When asked to sketch a quadratic graph, follow these sequential steps:

Determine if it is u-shaped or n-shaped based on the sign of $x^2$ .
Identify and label the $y$ -intercept $(0, c)$ .
Factorise or use the quadratic formula to find and label the roots.
Find the turning point (either midway between the roots or via completing the square) and label it.
Draw a single continuous smooth curve through your points.

Worked Example: Sketching a Quadratic Graph

Sketch the graph of $y = x^2 - 4x - 5$ .

Step 1: Determine the shape and label the $y$ -intercept. The coefficient of $x^2$ is positive ( $a=1$ ), so the graph is a u-shaped parabola. The $y$ -intercept is the constant term $c = - 5$ . Label the point on the -axis.

Step 2: Find and label the roots by setting $y = 0$ . $0 = x^2 - 4x - 5$ Factorise the quadratic: $0 = (x -$ The roots are and . Label and on the -axis.

Step 3: Find and label the turning point. The line of symmetry is exactly halfway between the roots: $x = \frac{5 + (-1)}{2} = 2$ . Substitute $x = 2$ back into the equation to find the -coordinate: . Label the minimum turning point at .

Step 4: Draw the curve. Draw a single continuous smooth u-shaped curve passing through $(-1, 0)$ , $(0, -5)$ , $(2, -9)$ , and $(5, 0)$ .

Recognising and Interpreting Cubic Graphs

Understanding how equations behave explains why some mathematical models rise, flatten out completely, and then rise again.

A cubic function ( $y = ax^3 + bx^2 + cx + d$ ) creates a distinctive "S-shape" or wiggle.
It can have up to two turning points (a local maximum and a ) or a single (like in ) where the curvature flattens and changes direction.

End Behavior of Cubic Graphs

End behavior describes the direction the graph travels as $x$ approaches positive or negative infinity.
If the $x^3$ term is positive ( $a > 0$ ), the graph starts bottom-left and ends top-right.
If the $x^3$ term is negative (), the graph starts top-left and ends bottom-right.

Worked Example: Sketching a Cubic Graph

Sketch the graph of $y = x^3 - 4x$ .

Step 1: Identify and label the $y$ -intercept. By setting $x = 0$ , $y = (0)^3 - 4(0) = 0$ . Label the point .

Step 2: Find and label the roots by setting $y = 0$ . $0 = x^3 - 4x$ Factorise by taking out $x$ : Factorise the difference of two squares: The roots are , , and . Label , , and on the -axis.

Step 3: Determine the end behavior and sketch the curve. The coefficient of $x^3$ is positive ( $a = 1$ ), so the graph starts bottom-left and ends top-right. Draw a smooth curve starting from the bottom-left, passing up through $(-2, 0)$ to a local maximum, coming down through $(0, 0)$ to a local minimum, and going back up through towards the top-right.

Students often fail to change the sign of p when identifying the vertex of a quadratic from the completed square form y = a(x + p)² + q; the vertex is at (-p, q), not (p, q).

When calculating a linear gradient from a graph, students often just count grid squares; always calculate using the axis values, as the x and y scales frequently differ.

When asked to sketch a quadratic or cubic graph, do not use a ruler to join the points; examiners will penalise you if the curve is not a single, continuous freehand line.

If an exam question asks for the 'line of symmetry', you must write it as a full equation (e.g., x = 3), not just a single number.

For Higher Tier questions asking for the gradient at a specific point on a curve, draw a straight tangent line touching that point and calculate the gradient of the tangent using two points on it.

An equation of the form y = mx + c that produces a straight-line graph with a constant rate of change.

The measure of the steepness of a line, calculated as the vertical change divided by the horizontal change.

A measure of how one quantity changes in relation to another, represented by the gradient of a linear graph.

The coordinate where x = 0 and a graph intersects the y-axis.

To draw a graph accurately on a grid using specific, calculated coordinates.

To draw an approximate diagram of a graph showing key features like intercepts, turning points, and general shape, without needing precise scale.

The symmetrical, U-shaped or N-shaped curve created by graphing a quadratic function.

The number multiplying the x² term in a quadratic equation, which determines whether the parabola is u-shaped or n-shaped.

The orientation of a parabola with a positive x² coefficient, resulting in a minimum turning point.

The orientation of a parabola with a negative x² coefficient, resulting in a maximum turning point.

The peak or trough of a curve where the gradient is zero and the graph changes direction.

The x-values where a graph crosses or touches the x-axis, representing the solutions when y = 0.

A vertical line passing through the turning point of a parabola that divides the curve into two identical mirror images.

A polynomial equation of degree 3 that produces an S-shaped graph.

A turning point on a curve that is higher than the points immediately around it.

A turning point on a curve that is lower than the points immediately around it.

A point on a curve where the curvature changes direction, often flattening out before continuing in the same overall vertical direction.

The general direction of a graph as the x-values approach extremely large positive or negative numbers.