Graphs and FunctionsAQA GCSE Maths: Revision notes
Section 1
How do linear functions work and what do gradient and y-intercept mean?
Linear functions follow the form y = mx + c, where:
- m is the gradient (slope) — how steep the line is
- c is the y-intercept — where the line crosses the y-axis
The gradient tells you how much y changes for every 1 unit change in x. A positive gradient slopes upward; negative slopes downward.
Parallel lines have the same gradient. For perpendicular lines (Higher Tier), if one line has gradient m, the perpendicular line has gradient −1/m (negative reciprocal).
Finding the equation of a line:
- If given two points: calculate gradient using m = (y₂ − y₁)/(x₂ − x₁), then substitute one point into y = mx + c to find c
- If given one point and gradient: substitute directly into y = mx + c and solve for c
Always rearrange into the form y = mx + c to identify gradient and y-intercept clearly.
Find the equation of the line through points (2, 5) and (4, 11). Gradient = (11 − 5)/(4 − 2) = 6/2 = 3. Using (2, 5): 5 = 3(2) + c, so c = −1. Equation: y = 3x − 1.
Examiners expect you to show the gradient formula clearly and identify m and c explicitly. State equations in the form y = mx + c unless asked otherwise.
Section 2
What are the key features of quadratic, cubic, reciprocal and exponential functions?
Quadratic functions (y = ax² + bx + c) produce parabola graphs:
- Turning point (vertex): the maximum or minimum point
- Roots (zeros): where the graph crosses the x-axis; solve by factorising or using the quadratic formula
- y-intercept: the value of c
- If a > 0, parabola opens upward; if a < 0, opens downward
Cubic functions (y = ax³ + bx² + cx + d):
- Have up to two turning points and up to three roots
- Change direction (increasing then decreasing or vice versa)
Reciprocal functions (y = k/x, where k is a constant):
- Produce two separate curves (hyperbolas) in opposite quadrants
- Never cross the axes; have asymptotes at x = 0 and y = 0
- Not defined at x = 0
Exponential functions (y = aˣ or y = a × bˣ):
- Always positive; curve never touches the x-axis
- Pass through (0, a) or relevant y-intercept
- Increase or decrease rapidly depending on the base
Trigonometric functions (Higher Tier): y = sin x, y = cos x, y = tan x have periodic patterns; sin and cos oscillate between −1 and 1, whilst tan has vertical asymptotes.
Students often forget that reciprocal functions are undefined at x = 0 and assume the curve must cross the axes. Remember: the asymptotes are the axes themselves.
For y = x² − 5x + 6, find roots: factorise to (x − 2)(x − 3) = 0, so roots are x = 2 and x = 3. Turning point at x = 5/2 = 2.5; substitute to find y = −0.25, so turning point is (2.5, −0.25).
Section 3
How do transformations affect function graphs?
Transformations (Higher Tier) change the position, shape or orientation of a graph:
| Transformation | Effect | Example |
|---|---|---|
| f(x) + a | Vertical shift up (or down if a is negative) by a units | y = x² becomes y = x² + 3 (shift up 3) |
| f(x + a) | Horizontal shift left (or right if a is negative) by a units | y = x² becomes y = (x + 2)² (shift left 2) |
| af(x) | Vertical stretch by factor a (if a > 1) or compression (if 0 < a < 1) | y = x² becomes y = 2x² (stretch by factor 2) |
| f(ax) | Horizontal compression by factor 1/a (if a > 1) or stretch (if 0 < a < 1) | y = x² becomes y = (2x)² (compress by factor 1/2) |
Key points to remember:
- Changes inside brackets (to x) affect horizontal direction opposite to the sign
- Changes outside brackets affect vertical direction in the same direction as the sign
- Always describe transformations in order: horizontal shifts, then stretches, then vertical shifts
- To reverse-engineer the transformation, identify key points (intercepts, turning points) on both graphs
When asked to describe a transformation, state it precisely: 'vertical shift up by 3 units' or 'horizontal shift left by 2 units' rather than vague descriptions. Examiners mark clarity highly.
Think of f(x + a) like a train moving left on a track: the number inside the bracket works 'backwards' from what you might expect.
Section 4
How are graphs used to solve equations and interpret real-world problems?
Solving equations graphically:
- To solve f(x) = k, find where the curve y = f(x) intersects the horizontal line y = k
- To solve f(x) = g(x), find where the two curves intersect
- Read the x-coordinates of intersection points from the graph; these are approximate solutions
Interpreting gradients as rates of change (Higher Tier):
- The gradient of a curve at a point represents the instantaneous rate of change
- To find it, draw a tangent to the curve at that point and calculate its gradient
- In real contexts (distance–time graphs, temperature change, etc.), this shows how quickly something is changing at that moment
Calculating areas under graphs (Higher Tier):
- Area under a curve represents quantities such as distance travelled or work done
- For irregular shapes, split into rectangles and triangles, or use the trapezium rule
- For straight lines, use simple geometry formulas
Non-standard functions in context:
- Always identify what the axes represent (e.g., time, distance, cost)
- Use the graph to answer questions about maximum/minimum values, when things happen, or approximate solutions
- Interpolate (read between known points) rather than extrapolate (guess beyond the graph)
To solve x² − 2x − 3 = 0 graphically, plot y = x² − 2x − 3 and find where it crosses the x-axis. Read off the x-coordinates: x = −1 and x = 3 are the solutions.
For rate of change questions, examiners always want to see the tangent drawn clearly and the gradient calculation shown step-by-step. Label points on the tangent with coordinates.
Section 5
What is direct and inverse proportion, and how are they represented?
Direct proportion (Higher Tier):
- Two variables are directly proportional if y = kx where k is a constant
- The graph is a straight line through the origin (0, 0)
- k is the constant of proportionality, equal to the gradient
- As x increases, y increases at a constant ratio
- Write as y ∝ x or use y = kx after finding k
Inverse proportion:
- Two variables are inversely proportional if y = k/x or xy = k where k is a constant
- The graph is a reciprocal curve (hyperbola) in the first and third quadrants
- As x increases, y decreases; the product xy is always constant
- Write as y ∝ 1/x
Finding the constant of proportionality:
- Substitute any known pair of values into the relationship
- Solve for k
- Write the full equation using this value of k
Graphical identification:
- Direct proportion: straight line through origin; recognisable by this passing-through-origin feature
- Inverse proportion: hyperbola with asymptotes at the axes; the two branches are in opposite quadrants
If y is inversely proportional to x and y = 8 when x = 2, find k: 8 = k/2, so k = 16. The equation is y = 16/x. Check: when x = 4, y = 4, confirming the relationship.
Students often confuse which relationship goes with which equation. Remember: direct is y = kx (linear, through origin); inverse is y = k/x (curved, asymptotes at axes).
Must Know
- Linear functions: y = mx + c where m is gradient and c is y-intercept. Parallel lines have equal gradients; perpendicular lines have gradients multiplying to −1.
- Roots, turning points and intercepts are found graphically by identifying where curves cross axes or reach maximum/minimum values.
- Key function types: quadratics (parabola with turning point), cubics (up to two turning points), reciprocal (hyperbola, asymptotes at axes), exponential (curve never touches x-axis), trigonometric (periodic oscillation).
- Transformations f(x) + a (vertical shift), f(x + a) (horizontal shift opposite to sign), af(x) (vertical stretch), f(ax) (horizontal compression) change graph position and shape.
- Solving equations graphically: find intersection points of curves or with horizontal lines; gradient of tangent represents rate of change; area under curves found using geometry or trapezium rule.
- Proportion: direct proportion is y = kx (straight line through origin), inverse is y = k/x (hyperbola); find k by substituting known values.
That's the notes covered.
Carry on to the next subtopic.