Chapter 4 of 10

📈 Functions & Graphs

5 min+15 XP

You need to sketch and interpret linear, quadratic, power, and exponential graphs. Key skills include finding gradients, y-intercepts, turning points, and using tangent lines to estimate gradients at a point. Graph questions appear in both papers and can carry heavy marks in Paper 2.

Understanding graphs is about connecting algebra to geometry. Every equation has a corresponding graph, and every graph tells a story about the equation. For example, the roots of an equation are the x-intercepts of the graph, the gradient tells you how steep the line is, and the turning point of a parabola tells you the maximum or minimum value. Mastering this connection is essential for scoring well.

Linear Graphs:

A linear equation produces a straight-line graph. The constant is the gradient (slope) of the line — it tells you how steep the line is and whether it slopes upward or downward. The constant is the y-intercept — where the line crosses the y-axis. Two key relationships: parallel lines have the same gradient, and perpendicular lines have gradients that multiply to give $-1$.

📈Linear Functions

Straight-line graphs. Key concepts: gradient, y-intercept, equation of a straight line.

Gradient-intercept form

Gradient between two points

Equation given point and gradient

Parallel lines (same gradient)

Perpendicular lines

x-intercept (set y = 0)

📝Worked Example 1: Finding the Equation of a Line

Question

Find the equation of the line passing through and .

y = 2x + 1 showing gradient, y-intercept, and x-intercept

📝Worked Example 2: Perpendicular Lines

Question

Line L1 has equation . Find the equation of line L2 perpendicular to L1 passing through .

L2: y = -x/3 + 4 showing gradient, y-intercept, and x-intercept

📝Worked Example 3: Finding x-intercept and y-intercept

Question

Find the x-intercept and y-intercept of the line .

y = 3x/2 - 6 showing gradient, y-intercept, and x-intercept

🧠Quick Check

A line passes through (1, 3) and (4, 9). What is its gradient?

D1/2

Quadratic Graphs:

Quadratic graphs are U-shaped parabolas. The coefficient a determines the shape: if , the parabola opens upward (minimum turning point). If , it opens downward (maximum turning point). The line of symmetry passes through the turning point at . You can also find the turning point by completing the square.

To sketch a quadratic graph, you need to find: (1) the y-intercept (set x = 0), (2) the x-intercepts (set y = 0 and solve), (3) the line of symmetry, and (4) the turning point. Plot these key features and draw a smooth curve through them.

🎯Quadratic Functions

Parabola graphs. Determine turning point, line of symmetry, and nature of the curve.

General form

Line of symmetry

Turning point (vertex)

Shape: a > 0 (minimum curve, U-shape)

Shape: a < 0 (maximum curve, inverted U)

Completed square form

📝Worked Example 4: Sketching a Quadratic

Question

Sketch . Find the turning point, y-intercept, and x-intercepts.

Quadratic graph showing vertex, intercepts, and axis of symmetry

📝Worked Example 5: Inverted Quadratic

Question

Sketch . Find the maximum point.

Quadratic graph showing vertex, intercepts, and axis of symmetry

🧠Quick Check

The line y = 3x - 1 is perpendicular to line L. What is the gradient of L?

B-3

C1/3

D-1/3

Exponential & Reciprocal Graphs

Exponential graphs ( where a > 0 and a is not 1) show rapid growth or decay. When , the graph shows exponential growth (increases rapidly as x increases). When , it shows exponential decay (decreases rapidly). All exponential graphs pass through (0, 1) because , and they have a horizontal asymptote at y = 0.

Reciprocal graphs () have two branches that form a hyperbola. They have two asymptotes: the x-axis (y = 0) and the y-axis (x = 0). The graph never touches either axis. When a > 0, the branches are in the first and third quadrants. When a < 0, they are in the second and fourth quadrants.

Exponential Graphs

Exponential growth (a > 1) and decay (0 < a < 1) both pass through (0, 1)

Reciprocal Graph: y = 3/x

Reciprocal graph y = 1/x with asymptotes at x = 0 and y = 0

🌊Other Graphs

Recognise and sketch power, exponential, and reciprocal graphs.

Cubic function

Reciprocal function

Exponential growth

Exponential decay

Power function

🔍Graph Interpretation

Use graphs to solve equations and estimate gradients of curves.

Solving equations graphically

Gradient of a curve at a point

Area under a graph (estimation)

📝Worked Example 6: Sketching an Exponential Growth Graph

Question

Sketch the graph of y = 2^x for -3 <= x <= 3. State the key features.

Exponential growth (a > 1) and decay (0 < a < 1) both pass through (0, 1)

📝Worked Example 6b: Exponential Decay

Question

The value of a car depreciates according to V = 25000(0.85)^t where V is the value in dollars and t is the number of years. (a) Find the initial value. (b) Find the value after 5 years. (c) Find how long until the value drops below $10000.

Exponential growth (a > 1) and decay (0 < a < 1) both pass through (0, 1)

📝Worked Example 7: Reciprocal Graph Problem

Question

The graph y = k/x passes through the point (2, 6). Find k and hence find the value of y when x = 4.

Reciprocal graph y = 1/x with asymptotes at x = 0 and y = 0

📝Worked Example 7b: Cubic Graph Features

Question

Sketch the graph of y = x^3 - 3x. Find the coordinates where the graph crosses the x-axis.

🧠Quick Check

Which graph passes through (0, 1)?

🌊Other Graphs

Recognise and sketch power, exponential, and reciprocal graphs.

Cubic function

Reciprocal function

Exponential growth

Exponential decay

Power function

Gradient of a Curve at a Point

Unlike a straight line, the gradient of a curve changes at every point. To find the gradient at a specific point, you draw a tangent line (a straight line that just touches the curve at that point) and then calculate the gradient of the tangent. This is an estimation — you choose two points on the tangent line that are far apart for a more accurate result.

🔍Graph Interpretation

Use graphs to solve equations and estimate gradients of curves.

Solving equations graphically

Gradient of a curve at a point

Area under a graph (estimation)

📝Worked Example 6: Interpreting a Distance-Time Graph

Question

A car travels for 2 hours at constant speed (120 km covered), stops for 30 minutes, then returns home at 80 km/h. Describe the distance-time graph and find the total time.

📝Worked Example 7: Solving Equations Graphically

Question

The graph of is drawn. Use the graph to solve and .

Quadratic graph showing vertex, intercepts, and axis of symmetry

🧠Quick Check

For , what is the x-coordinate of the turning point?

B-2

D-4

⚠️Common Mistakes — Graphs

Common Mistake

Gradient of a horizontal line is undefined.

Correct

Horizontal line: gradient . Vertical line: gradient is undefined.

Why: Horizontal: rise = 0, so m = 0/run = 0. Vertical: run = 0, so m = rise/0 = undefined. Students often swap these.

Common Mistake

has positive gradient because 5 is positive.

Correct

Gradient (coefficient of ). The -intercept is 5.

Why: In y = mx + c, the gradient is m (coefficient of x), not c. Do not confuse gradient with y-intercept.

Common Mistake

Exterior angle of regular hexagon

Correct

Exterior angle of regular hexagon

Why: Sum of exterior angles of ANY convex polygon is 360 degrees (not 180). Each exterior angle = 360/n.

Speed-Time Graphs: Area and Gradient

Speed-time graphs are a favourite Paper 2 topic. The two key relationships are: (1) the gradient of a speed-time graph gives the acceleration, and (2) the area under a speed-time graph gives the total distance travelled. These two facts allow you to extract all the information from any speed-time graph.

Feature	Distance-Time Graph	Speed-Time Graph
Gradient represents	Speed	Acceleration
Area under graph	Not meaningful	Total distance
Horizontal line means	Stationary (not moving)	Constant speed
Straight line going up	Constant speed	Constant acceleration
Curve	Changing speed	Changing acceleration

📝Worked Example 12b: Speed-Time Graph — Finding Distance

Question

A train accelerates uniformly from rest to 30 m/s in 20 seconds, travels at constant speed for 40 seconds, then decelerates uniformly to rest in 15 seconds. Find the total distance.

📝Worked Example 12c: Speed-Time Trapezoid

Question

A car accelerates from 10 m/s to 25 m/s in 6 seconds. Find (a) the acceleration and (b) the distance covered during this acceleration.

🧠Quick Check

On a speed-time graph, a horizontal line at m/s from to seconds. What is the distance?

A100 m

B200 m

C300 m

D400 m

Graph Transformations

Graph transformations describe how a base graph changes when you modify the equation. Understanding these transformations helps you sketch new graphs quickly from a known graph. There are four main transformations at O-Level.

Transformation	Effect on Graph	Example
	Translate UP by units	: shifted up 3
	Translate DOWN by units	: shifted down 2
	Translate RIGHT by units	: shifted right 3
	Translate LEFT by units	: shifted left 2
	Reflect in the x-axis	: flipped upside down
	Reflect in the y-axis	(symmetric, same graph)
	Stretch vertically by factor	: stretched 3x taller

📝Worked Example 12d: Describing a Graph Transformation

Question

The graph of is transformed to give . Describe the transformation.

Quadratic graph showing vertex, intercepts, and axis of symmetry

📝Worked Example 12e: Applying a Transformation

Question

The curve passes through (3, 7). Find the coordinates of the corresponding point on (a) (b) (c) .

Quadratic graph showing vertex, intercepts, and axis of symmetry

🧠Quick Check

y = f(x) is transformed to y = f(x + 3). The point (5, 2) on y = f(x) maps to:

A(8, 2)

B(2, 2)

C(5, 5)

D(5, -1)

Graph Sketching Checklist

When asked to sketch a graph, follow this systematic approach to ensure you include all required features. Marks are awarded for each feature, so a methodical approach maximises your score even if your curve is not perfectly smooth.

How to Sketch Any Graph

Step 1: Identify the graph type

Linear (straight line), quadratic (parabola), reciprocal (hyperbola), exponential, or cubic? This determines the basic shape.

Step 2: Find the y-intercept

Set and calculate . Plot this point. For , the y-intercept is .

Step 3: Find the x-intercepts

Set and solve. For quadratics, factorise or use the formula. These are where the curve crosses the x-axis.

Step 4: Find turning points/asymptotes

For quadratics: use . For reciprocals: asymptotes at and . For exponentials: asymptote at .

Step 5: Plot key points and draw

Plot at least 5 points for curves. Connect with a smooth curve (not straight-line segments). Label the axes and key features.

📝Worked Example 8: Completing the Square to Find Turning Point

Question

Find the turning point of by completing the square.

Quadratic graph showing vertex, intercepts, and axis of symmetry

📝Worked Example 9: Using a Graph to Solve an Equation

Question

Given the graph of , use the graph to solve .

Quadratic graph showing vertex, intercepts, and axis of symmetry

Important: Sketch vs Draw vs Plot

Sketch means draw a rough version showing the correct shape and key features (intercepts, turning points, asymptotes) — it does not need to be perfectly accurate. Draw means use a ruler for straight lines and draw accurately. Plot means calculate exact points and mark them precisely on the graph paper.

🧠Quick Check

The graph of passes through which of these points?

A(0, 0)

B(0, 2)

C(0, 1)

D(2, 0)

Gradient of a Curve at a Point

The gradient of a straight line is constant, but the gradient of a curve changes at every point. To find the gradient of a curve at a specific point, you draw a tangent line at that point and then calculate the gradient of the tangent line. The tangent is the straight line that just touches the curve at that point without crossing it.

📝Worked Example 10: Drawing a Tangent and Finding Gradient

Question

The graph of y = x^2 - 4x + 3 is drawn. Estimate the gradient at the point (3, 0) by drawing a tangent.

Quadratic graph showing vertex, intercepts, and axis of symmetry

📝Worked Example 11: Distance-Time Graph

Question

A car travels for 2 hours. For the first hour, it covers 60 km at constant speed. It stops for 30 minutes. Then it returns to the start at constant speed.

📝Worked Example 12: Speed-Time Graph

Question

A speed-time graph shows: speed increases from 0 to 20 m/s in 5 seconds (uniform acceleration), constant speed of 20 m/s for 10 seconds, then decelerates to 0 in 5 seconds. Find (a) the acceleration, (b) the total distance.

Graph Interpretation Summary

For travel graphs, remember these key interpretations:

Distance-time graph: Gradient = speed. Area has no meaning.
Speed-time graph: Gradient = acceleration. Area = distance.
Horizontal line: On d-t graph = stationary. On s-t graph = constant speed.
Steeper line: On d-t graph = faster. On s-t graph = greater acceleration.
Negative gradient: On d-t graph = returning. On s-t graph = decelerating.

🧠Quick Check

On a speed-time graph, what does the area under the graph represent?

ASpeed

BAcceleration

CDistance

DTime

🎯Key Takeaway

For graph questions, always find key features: intercepts, turning points, asymptotes. Know the shapes of all graph types (linear, quadratic, reciprocal, exponential). Draw smooth curves, not straight-line segments. To find the gradient of a curve at a point, draw a tangent line. To solve equations graphically, rearrange to use the given curve. For travel graphs: d-t gradient = speed, s-t gradient = acceleration, s-t area = distance.