Graphs of Linear and Quadratic Functions

What is a Graph Sketch?

When giving someone directions, you might draw a quick map showing key landmarks rather than a perfectly scaled architect's blueprint. In GCSE Maths, a sketch is exactly that. It is a depiction showing the general shape and important features of a graph without needing a perfect scale or plotted grid. You do not need to calculate every single coordinate. However, you must explicitly label the crucial "landmarks": the intercepts, the turning points, and the correct overall shape.

Sketching Linear Graphs

Every straight line follows the general equation $y = mx + c$ . The gradient ( $m$ ) tells you the steepness and direction of the line. A positive gradient slopes upwards from left to right, while a negative gradient slopes downwards. The $y$ -intercept ( $c$ ) is the coordinate where the line crosses the $y$ -axis (the point $(0, c)$ ).

To sketch a linear graph:

Step 1: Identify the $y$ -intercept from the value of $c$ .
Step 2: Find the $x$ -intercept by setting $y = 0$ and solving for $x$ .
Step 3: Draw a straight line through these two points.
Step 4: Always use a ruler to draw linear graphs — AQA mark schemes strictly penalise freehand straight lines. Extend your line to the full edges of the provided axes.

Worked Example: Sketching a Linear Graph

Sketch the graph of $y = 2x - 4$ , labelling the intercepts and demonstrating the gradient.

Step 1: Identify the $y$ -intercept.

The equation is in the form $y = mx + c$ , where $c = -4$ .
The $y$ -intercept is $(0, -4)$ .

Step 2: Find the $x$ -intercept.

Set $y = 0$ : $0 = 2x - 4$ .
Solve for $x$ : $2x = 4$ , so $x = 2$ .
The $x$ -intercept is $(2, 0)$ .

Step 3: Draw the line and label the gradient.

Plot the coordinates $(0, -4)$ and $(2, 0)$ and join them using a ruler, extending the line across the axes.
The gradient is $m = 2$ . To explicitly label or demonstrate the gradient on the sketch, draw a small right-angled triangle along the line showing that for every $1$ unit across (horizontally), the line moves up $2$ units (vertically).

Sketching Quadratic Graphs

A quadratic function ( $y = ax^2 + bx + c$ ) forms a perfectly symmetrical curve called a parabola. The shape of this curve depends entirely on the coefficient of the $x^2$ term (the value of $a$ ).

If $a > 0$ (positive), the graph is a "U-shaped" curve with a minimum point.
If $a < 0$ (negative), the graph is an "n-shaped" curve with a maximum point.

When sketching a quadratic, you must label four key features:

The general shape (U-shaped or n-shaped).
The $y$ -intercept (found by setting $x = 0$ , which gives the value of $c$ ).
The roots (the $x$ -intercepts where $y = 0$ ).
The turning point (also called the vertex), where the graph changes direction.

Unlike linear graphs, a parabola must NOT be drawn with a ruler. You must draw a smooth freehand curve.

Worked Example: Sketching a Quadratic

Sketch the graph of $y = x^2 - 6x + 5$ , labelling all intercepts and the turning point.

Step 1: Identify the shape and $y$ -intercept.

The $x^2$ term is positive ( $a = 1$ ), so it is a U-shaped parabola.
Substitute $x = 0$ : $y = 0^2 - 6(0) + 5 = 5$ . The $y$ -intercept is $(0, 5)$ .

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

Set $y = 0$ and factorise: $x^2 - 6x + 5 = 0$ .
$(x - 1)(x - 5) = 0$ .
The roots are at $x = 1$ and $x = 5$ . Label the points $(1, 0)$ and $(5, 0)$ .

Step 3: Find the turning point.

Parabolas are perfectly symmetrical. The $x$ -coordinate of the turning point is exactly halfway between the roots ( $1$ and $5$ ). The midpoint is $x = 3$ .
Substitute $x = 3$ back into the original equation to find the $y$ -coordinate: $y = 3^2 - 6(3) + 5 = 9 - 18 + 5 = -4$ .
The turning point is at $(3, -4)$ .

(Higher Tier note: You can also find the turning point algebraically by completing the square to get $y = (x - 3)^2 - 4$ , which instantly gives the vertex $(3, -4)$ ).

Interpreting Graphs in Context

Graph features often represent real-world values. When an exam question asks you to "interpret" a graph, you must connect the mathematics back to the story in the question.

$y$ -intercept: Represents the "initial value" when time, distance, or usage is zero. For example, on a plumber's cost graph ( $C = 30h + 40$ ), the $y$ -intercept ( $40$ ) represents the fixed call-out fee (£40).
Gradient: Represents a "rate of change". In the plumber example, the gradient ( $30$ ) means the cost increases by £30 per hour.
Roots: Often represent when a physical object hits the ground (height = $0$ ) or when a business breaks even (profit = $0$ ).
Turning Point: Represents the absolute maximum (e.g., the peak height of a thrown ball) or absolute minimum (e.g., the lowest cost point of a production process).

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Exam Tips

Common Mistake
Students often lose marks by joining the points of a quadratic graph with a ruler. AQA strictly requires a smooth freehand curve for parabolas.
Common Mistake
When reading the turning point from a completed square format like $y = (x + 4)^2 + 3$ , students forget to reverse the sign inside the bracket. The $x$ -coordinate is $-4$ , not $4$ .
3
When sketching linear graphs, AQA examiners will penalise you if you do not use a ruler or straight edge.
4
In an exam, if a straight line passes exactly through the origin ( $y = 2x$ ), label $(0,0)$ and plot one other point (like substituting $x=1$ ) to clearly show the gradient's steepness.
5
When a question asks you to 'interpret' a gradient or intercept, do not just define the math term — you must state what it means in the real-world context provided (e.g., 'the starting temperature' or 'the speed').

Key Terms(9)

Sketch: A depiction showing the general shape and important features (like intercepts and turning points) of a graph without needing a perfect scale or plotted grid.
Gradient: A measure of the steepness and direction of a line, calculated as the vertical change for every $1$ unit of horizontal change.
y-intercept: The coordinate where a graph intersects the $y$ -axis, representing the value of $y$ when $x = 0$ .
x-intercept: The coordinate where a graph intersects the $x$ -axis, representing the value of $x$ when $y = 0$ .
Parabola: The distinctive U-shaped or n-shaped perfectly symmetrical curve formed by a quadratic function.
Coefficient: The numerical factor multiplying a variable in an algebraic term (e.g., the $a$ in $ax^2$ ).
Roots: The $x$ -values where a graph crosses the $x$ -axis, calculated by setting the equation's $y$ -value to zero.
Turning point: The coordinate where a curve changes direction and the gradient becomes exactly zero.
Vertex: Another mathematical term for the turning point (the absolute maximum or minimum) of a parabola.

Previous Subtopic: Quadratic Graph PropertiesQuadratic Graph Properties Next NoteGraphs of Cubic and Reciprocal Functions

Back to Function Graphs

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Key Terms(9)

Sketch: A depiction showing the general shape and important features (like intercepts and turning points) of a graph without needing a perfect scale or plotted grid.
Gradient: A measure of the steepness and direction of a line, calculated as the vertical change for every $1$ unit of horizontal change.
y-intercept: The coordinate where a graph intersects the $y$ -axis, representing the value of $y$ when $x = 0$ .
x-intercept: The coordinate where a graph intersects the $x$ -axis, representing the value of $x$ when $y = 0$ .
Parabola: The distinctive U-shaped or n-shaped perfectly symmetrical curve formed by a quadratic function.
Coefficient: The numerical factor multiplying a variable in an algebraic term (e.g., the $a$ in $ax^2$ ).
Roots: The $x$ -values where a graph crosses the $x$ -axis, calculated by setting the equation's $y$ -value to zero.
Turning point: The coordinate where a curve changes direction and the gradient becomes exactly zero.
Vertex: Another mathematical term for the turning point (the absolute maximum or minimum) of a parabola.

Graphs of Linear and Quadratic Functions

What is a Graph Sketch?

Sketching Linear Graphs

To sketch a linear graph:

Step 1: Identify the $y$ -intercept from the value of $c$ .
Step 2: Find the $x$ -intercept by setting $y = 0$ and solving for $x$ .
Step 3: Draw a straight line through these two points.
Step 4: Always use a ruler to draw linear graphs — AQA mark schemes strictly penalise freehand straight lines. Extend your line to the full edges of the provided axes.

Worked Example: Sketching a Linear Graph

Sketch the graph of $y = 2x - 4$ , labelling the intercepts and demonstrating the gradient.

Step 1: Identify the $y$ -intercept.

The equation is in the form $y = mx + c$ , where $c = -4$ .
The $y$ -intercept is $(0, -4)$ .

Step 2: Find the $x$ -intercept.

Set $y = 0$ : $0 = 2x - 4$ .
Solve for $x$ : $2x = 4$ , so $x = 2$ .
The $x$ -intercept is $(2, 0)$ .

Step 3: Draw the line and label the gradient.

Plot the coordinates $(0, -4)$ and $(2, 0)$ and join them using a ruler, extending the line across the axes.
The gradient is $m = 2$ . To explicitly label or demonstrate the gradient on the sketch, draw a small right-angled triangle along the line showing that for every $1$ unit across (horizontally), the line moves up $2$ units (vertically).

Sketching Quadratic Graphs

If $a > 0$ (positive), the graph is a "U-shaped" curve with a minimum point.
If $a < 0$ (negative), the graph is an "n-shaped" curve with a maximum point.

When sketching a quadratic, you must label four key features:

The general shape (U-shaped or n-shaped).
The $y$ -intercept (found by setting $x = 0$ , which gives the value of $c$ ).
The roots (the $x$ -intercepts where $y = 0$ ).
The turning point (also called the vertex), where the graph changes direction.

Unlike linear graphs, a parabola must NOT be drawn with a ruler. You must draw a smooth freehand curve.

Worked Example: Sketching a Quadratic

Sketch the graph of $y = x^2 - 6x + 5$ , labelling all intercepts and the turning point.

Step 1: Identify the shape and $y$ -intercept.

The $x^2$ term is positive ( $a = 1$ ), so it is a U-shaped parabola.
Substitute $x = 0$ : $y = 0^2 - 6(0) + 5 = 5$ . The $y$ -intercept is $(0, 5)$ .

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

Set $y = 0$ and factorise: $x^2 - 6x + 5 = 0$ .
$(x - 1)(x - 5) = 0$ .
The roots are at $x = 1$ and $x = 5$ . Label the points $(1, 0)$ and $(5, 0)$ .

Step 3: Find the turning point.

Parabolas are perfectly symmetrical. The $x$ -coordinate of the turning point is exactly halfway between the roots ( $1$ and $5$ ). The midpoint is $x = 3$ .
Substitute $x = 3$ back into the original equation to find the $y$ -coordinate: $y = 3^2 - 6(3) + 5 = 9 - 18 + 5 = -4$ .
The turning point is at $(3, -4)$ .

(Higher Tier note: You can also find the turning point algebraically by completing the square to get $y = (x - 3)^2 - 4$ , which instantly gives the vertex $(3, -4)$ ).

Interpreting Graphs in Context

Graph features often represent real-world values. When an exam question asks you to "interpret" a graph, you must connect the mathematics back to the story in the question.

$y$ -intercept: Represents the "initial value" when time, distance, or usage is zero. For example, on a plumber's cost graph ( $C = 30h + 40$ ), the $y$ -intercept ( $40$ ) represents the fixed call-out fee (£40).
Gradient: Represents a "rate of change". In the plumber example, the gradient ( $30$ ) means the cost increases by £30 per hour.
Roots: Often represent when a physical object hits the ground (height = $0$ ) or when a business breaks even (profit = $0$ ).
Turning Point: Represents the absolute maximum (e.g., the peak height of a thrown ball) or absolute minimum (e.g., the lowest cost point of a production process).

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often lose marks by joining the points of a quadratic graph with a ruler. AQA strictly requires a smooth freehand curve for parabolas.
Common Mistake
When reading the turning point from a completed square format like $y = (x + 4)^2 + 3$ , students forget to reverse the sign inside the bracket. The $x$ -coordinate is $-4$ , not $4$ .
3
When sketching linear graphs, AQA examiners will penalise you if you do not use a ruler or straight edge.
4
In an exam, if a straight line passes exactly through the origin ( $y = 2x$ ), label $(0,0)$ and plot one other point (like substituting $x=1$ ) to clearly show the gradient's steepness.
5
When a question asks you to 'interpret' a gradient or intercept, do not just define the math term — you must state what it means in the real-world context provided (e.g., 'the starting temperature' or 'the speed').

Key Terms(9)

Sketch: A depiction showing the general shape and important features (like intercepts and turning points) of a graph without needing a perfect scale or plotted grid.
Gradient: A measure of the steepness and direction of a line, calculated as the vertical change for every $1$ unit of horizontal change.
y-intercept: The coordinate where a graph intersects the $y$ -axis, representing the value of $y$ when $x = 0$ .
x-intercept: The coordinate where a graph intersects the $x$ -axis, representing the value of $x$ when $y = 0$ .
Parabola: The distinctive U-shaped or n-shaped perfectly symmetrical curve formed by a quadratic function.
Coefficient: The numerical factor multiplying a variable in an algebraic term (e.g., the $a$ in $ax^2$ ).
Roots: The $x$ -values where a graph crosses the $x$ -axis, calculated by setting the equation's $y$ -value to zero.
Turning point: The coordinate where a curve changes direction and the gradient becomes exactly zero.
Vertex: Another mathematical term for the turning point (the absolute maximum or minimum) of a parabola.

Previous Subtopic: Quadratic Graph PropertiesQuadratic Graph Properties Next NoteGraphs of Cubic and Reciprocal Functions

Back to Function Graphs

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(9)

Sketch: A depiction showing the general shape and important features (like intercepts and turning points) of a graph without needing a perfect scale or plotted grid.
Gradient: A measure of the steepness and direction of a line, calculated as the vertical change for every $1$ unit of horizontal change.
y-intercept: The coordinate where a graph intersects the $y$ -axis, representing the value of $y$ when $x = 0$ .
x-intercept: The coordinate where a graph intersects the $x$ -axis, representing the value of $x$ when $y = 0$ .
Parabola: The distinctive U-shaped or n-shaped perfectly symmetrical curve formed by a quadratic function.
Coefficient: The numerical factor multiplying a variable in an algebraic term (e.g., the $a$ in $ax^2$ ).
Roots: The $x$ -values where a graph crosses the $x$ -axis, calculated by setting the equation's $y$ -value to zero.
Turning point: The coordinate where a curve changes direction and the gradient becomes exactly zero.
Vertex: Another mathematical term for the turning point (the absolute maximum or minimum) of a parabola.