GradeGen.AI

Sketching Linear Graphs

Recognition of functions starts with the highest power of $x$ . In a linear function, the highest power of $x$ is 1. Every time you walk up a hill or pay a fixed standing charge on a utility bill, you are experiencing a linear relationship. The graph of a linear function is always a perfectly straight line.

The standard equation for a linear graph is written as:

y = mx + c

In this format, represents the (the steepness of the line), and represents the (the point where the line crosses the -axis). A positive gradient slopes upwards from left to right, while a negative gradient slopes downwards. The , or -intercept, is the point where the line crosses the -axis, which occurs when .

If an equation is given in a different format, such as $ax + by = k$ , you must first rearrange it into $y = mx + c$ before identifying the gradient and $y$ -intercept.

Worked Example: Linear Sketch

Sketch the graph of $y = -2x + 6$ , showing all intercepts.

Step 1: Identify the $y$ -intercept.

The value of $c$ is $6$ , so mark the point $(0, 6)$ on the $y$ -axis.

Step 2: Find the $x$ -intercept (root).

Set $y = 0$ to find where it crosses the $x$ -axis.
$0 = -2x + 6$
$2x = 6$

Step 3: Sketch the line.

Draw a straight line passing through $(0, 6)$ and $(3, 0)$ . Ensure the axes are labelled and the line clearly shows a negative gradient.

Quadratic Graphs: Parabolas and Turning Points

A quadratic function is recognised by having $x^2$ as its highest power. Throw a ball through the air, and its path traces a perfect, symmetrical curve. This curve is called a parabola, and it is the characteristic shape of any quadratic function in the form $y = ax^2 + bx + c$ .

The shape of the parabola depends on the $x^2$ coefficient. If $a > 0$ , the graph is U-shaped and has a minimum point. If $a < 0$ , the graph is n-shaped and has a maximum point. This highest or lowest spot is known as the turning point (or vertex). The simplest quadratic is $y = x^2$ , which has its turning point at the origin .

Every quadratic graph has a vertical line of symmetry passing straight through its turning point. The curve will cross the $y$ -axis exactly once at $(0, c)$ , and it may have zero, one, or two roots ( $x$ -intercepts) depending on where it crosses the $x$ -axis.

Worked Example: Quadratic Sketch

Sketch the graph of $y = x^2 - 2x - 8$ , showing all key features.

Step 1: Find the $y$ -intercept.

Substitute $x = 0$ into the equation: $y = -8$ .
The $y$ -intercept is $(0, -8)$ .

Step 2: Find the roots ( $x$ -intercepts).

Set $y = 0$ and factorise the quadratic.
$x^2 - 2x - 8 = 0$
$(x - 4)$

Step 3: Find the turning point by completing the square.

$y = (x - 1)^2 - 1^2 - 8$
$y = ($

Step 4: Sketch the parabola.

Draw a smooth, symmetrical U-shaped curve.
Mark the intercepts at $(-2, 0)$ , $(4, 0)$ , and $(0, -8)$ .
Label the turning point $(1, -9)$ . Ensure the curve is smooth at the bottom rather than a "V" shape.

Cubic Graphs: The S-Shape

A cubic function is one where the highest power of $x$ is 3. The standard cubic graph $y = x^3$ has a continuous, smooth "S-shape" (or wiggle). Crucially, the simple $y = x^3$ graph has zero turning points; it never changes its overall upward direction.

Instead of a turning point, $y = x^3$ passes exactly through the origin $(0, 0)$ , which acts as a point of inflection. At this exact coordinate, the gradient is momentarily zero (the curve flattens out) before the graph continues to rise.

Worked Example: Cubic Sketch

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

Step 1: Mark the point of inflection.

Mark the origin $(0, 0)$ . This is the center of the "wiggle" where the graph flattens.

Step 2: Identify reference points for curvature.

Calculate coordinates to ensure the curve is drawn at the correct steepness:
- For $x = 1$ , $y = 1^3 = 1$ . Plot $(1, 1)$ .

Step 3: Sketch the curve.

Draw a single, smooth curve starting in the 3rd quadrant (bottom-left), passing through $(-1, -1)$ , flattening at $(0, 0)$ , and rising steeply through $(1, 1)$ into the 1st quadrant (top-right).

Reciprocal Graphs and Asymptotes

A reciprocal function takes the form $y = \frac{a}{x}$ . Its graph produces a hyperbola, a smooth curve split into two completely separate, symmetrical branches. The most important feature of a reciprocal graph is its asymptotes. An asymptote is a line that the curve approaches infinitely closely but never touches or crosses.

Worked Example: Reciprocal Sketch

Sketch the graph of $y = \frac{1}{x}$ .

Step 1: Identify the asymptotes.

The function is undefined at $x = 0$ , and $y$ can never be $0$ .
The asymptotes are the $y$ -axis ( $x = 0$ ) and the -axis ().

Step 2: Plot reference points.

Since $a=1$ is positive, the branches are in the 1st and 3rd quadrants.
For $x = 1$ , $y = 1$ . Plot $(1, 1)$ .

Step 3: Sketch the hyperbola.

Draw two smooth curves. One branch must start near the positive $y$ -axis, pass through $(1, 1)$ , and curve toward the positive $x$ -axis. The other branch is its reflection through the origin in the 3rd quadrant. Ensure the curve never touches the axes.

Students often confuse signs when extracting the turning point from the completed square form—remember that for $y = (x + p)^2 + q$ , the turning point is at $(-p, q)$ , not $(p, q)$ . For example, $y = (x + 2)^2 + 3$ has a turning point at $(-2, 3)$ .

When sketching a reciprocal graph like $y = \frac{1}{x}$ , you must leave a clear, visible gap between your curve and the axes to show you understand they are asymptotes; touching the axis will lose you marks.

Always rearrange linear equations into the standard $y = mx + c$ format before stating the gradient. For example, in $2y = 4x + 1$ , the gradient is , not .

In OCR mark schemes, cubic and reciprocal curves must be drawn smoothly with a single continuous motion. Do not use a ruler or draw 'zigzag' lines for $y = x^3$ or $y = \frac{1}{x}$ .

A 'sketch' does not require graph paper or a perfect scale, but you MUST label the $x$ and $y$ axes, plot intercepts in the correct relative positions, and show the correct general shape.

Remember that the simple cubic $y = x^3$ has zero turning points; it is always increasing from left to right. Do not draw it with 'humps' like a quadratic.

A measure of the steepness of a line; the change in the $y$ -coordinate for every unit increase in the $x$ -coordinate.

The point where a graph crosses the $y$ -axis, occurring when $x = 0$ .

An $x$ -intercept of a graph, found by setting the function $y = 0$ .

The characteristic U-shaped or n-shaped curve produced by a quadratic function.

The maximum or minimum point on a curve where the graph changes direction.

A polynomial function where the highest power of $x$ is 3, such as $y = x^3$ .

A point where the curvature of a graph changes; in $y = x^3$ , this is the origin where the gradient is momentarily zero.

A function of the form $y = a/x$ (where $a \neq 0$ ), which produces a two-part curve called a hyperbola.

The smooth, two-part symmetrical curve characteristic of a reciprocal graph.

A line that a curve approaches infinitely closely but never reaches or crosses.