Learning Objectives
- Construct tables of values and draw graphs of polynomial, reciprocal and exponential functions.
- Recognise and sketch linear, quadratic, cubic, reciprocal and exponential graphs.
- Identify roots, turning points, symmetry and asymptotes where applicable.
- Solve equations graphically using roots and intersections.
- Estimate the gradient of a curve by drawing a tangent.
Key Terms
- Function graph
- A graph showing the output produced for each input in a rule.
- Root
- An x-value where the graph has y=0.
- Intersection
- A point shared by two graphs.
- Turning point
- A local maximum or minimum where the direction of a curve changes.
- Asymptote
- A line that a curve approaches but does not meet in the usual graph region.
- Tangent
- A straight line that touches a curve at a point and has the same local gradient.
- Graphical solution
- An approximate solution obtained by reading a graph.
What This Chapter Covers
- Tables and accurate curve drawing
- Standard shapes of important functions
- Roots, turning points, symmetry and asymptotes
- Solving equations by intersections
- Gradients of curves using tangents
Constructing A Table Of Values
Choose or use the specified x-values, substitute accurately, and record outputs to a suitable accuracy. Negative inputs must be placed in brackets, especially for even and odd powers. If the table contains a missing value, calculate it before plotting.
Worked Example: Cubic Table
Question: Find y for y=x^3-2x when x=-2,-1,0,1,2.
- x=-2: y=-8+4=-4.
- x=-1: y=-1+2=1.
- x=0: y=0.
- x=1: y=1-2=-1.
- x=2: y=8-4=4.
Answer: Points: (-2,-4),(-1,1),(0,0),(1,-1),(2,4).
Drawing Curves Accurately
Plot points as small crosses and join them with one smooth curve. A curve should pass close to all accurate points without artificial corners. Label axes and scales. Reciprocal graphs may have separate branches; never join branches across an asymptote.
Recognising Standard Graph Shapes
| Function Type | Typical Form | Important Features |
|---|---|---|
| Linear | y=mx+c | Straight line; constant gradient |
| Quadratic | y=ax^2+bx+c | Parabola; one turning point; vertical symmetry |
| Cubic | y=ax^3+b or related cubic | S-shaped or may have two turning points; often rotational behaviour |
| Reciprocal | y=a/x+b | Two branches; vertical and horizontal asymptotes |
| Exponential | y=ar^x+b | Growth if r>1, decay if 0<r<1; horizontal asymptote |
Quadratic Features
Roots occur where the graph crosses or touches the x-axis. The turning point is a minimum when the coefficient of x^2 is positive and a maximum when it is negative. The axis of symmetry passes vertically through the turning point.
Cubic Features
A simple graph y=x^3 passes through the origin and has rotational symmetry about the origin. Translations shift the centre of this behaviour. More general cubics may have two turning points or none. The graph can have one, two repeated, or three real roots depending on where it meets the x-axis.
Reciprocal Features
For y=\frac{a}{x}+b, the vertical asymptote is x=0 and the horizontal asymptote is y=b. The curve approaches the vertical asymptote but cannot meet it because division by zero is undefined.
Worked Example: Reciprocal Asymptotes
Question: State the asymptotes of y=\frac{3}{x}-4.
- The denominator is zero at x=0, giving the vertical asymptote.
- As |x| becomes large, the fraction tends to zero, so y tends to -4.
Answer: x=0 and y=-4.
Exponential Graphs
For y=ar^x+c, the graph approaches y=c as a horizontal asymptote in one direction. It does not cross that asymptote when a\ne0. Exponential growth becomes increasingly steep; exponential decay falls rapidly and then levels off.
Worked Example: Identify Growth Or Decay
Question: Describe y=5(0.6)^x+2.
- The base 0.6 lies between 0 and 1, so it is decay.
- The vertical translation +2 gives horizontal asymptote y=2.
- At x=0, y=5+2=7.
Answer: A decreasing exponential curve through (0,7) approaching y=2.
Solving An Equation From A Single Graph
If y=f(x) is already drawn, solutions of f(x)=0 are the roots. Solutions of f(x)=k are found by drawing y=k and reading intersections.
Worked Example: Horizontal Line Method
Question: The graph of y=x^3-3x is drawn. Explain how to solve x^3-3x=2.
- Draw the horizontal line y=2.
- Read the x-coordinates where the line meets the cubic.
- Quote answers to the precision supported by the graph.
Answer: The x-coordinates of the intersections are the graphical solutions.
Solving By Intersecting Two Graphs
To solve f(x)=g(x), draw both y=f(x) and y=g(x). The x-coordinates of their intersections satisfy the equation. If one graph is already supplied, rearrange the equation so that the new line or curve is simple to draw.
Worked Example: Line And Curve
Question: Explain how to solve x^2-4x-1=2x+3 graphically.
- Draw y=x^2-4x-1.
- Draw y=2x+3 on the same axes.
- Read the x-coordinates of intersection.
Answer: The intersections give the approximate roots of x^2-6x-4=0.
Transforming An Equation For A Supplied Graph
Suppose the graph of y=x^2+1/x is supplied and the equation to solve is x^2+1/x=4-x. Draw the straight line y=4-x. This is more efficient than redrawing the curve.
Estimating The Gradient Of A Curve
The gradient changes from point to point on a curve. Draw a tangent at the required point, choose two well-separated points on the tangent, and calculate rise over run. The points used need not lie on the curve; they must lie on the tangent.
Tangent Method
- Mark the point clearly.
- Place a ruler so the line touches the curve locally with a balanced gap on each side.
- Draw a long tangent.
- Choose two clear points far apart on the tangent.
- Calculate \Delta y/\Delta x, including units if the graph is practical.
Sketching Curves
A sketch is not required to be to scale, but it must show the correct quadrants, intercepts, turning points, asymptotes and symmetry. It should interact with axes and asymptotes correctly. Label important coordinates or values.
Examination Guidance
- Use a smooth curve and never join curve points with a zigzag of straight segments.
- For graphical solutions, state that answers are approximate and quote a sensible number of decimal places.
- When drawing a second graph, use the same scale and label it clearly.
- Do not join separate branches across a vertical asymptote.
- For tangent gradients, use a large triangle on the tangent to reduce reading error.
Common Mistakes
- Treating a graphical answer as exact.
- Finding y-coordinates instead of x-coordinates at intersections.
- Drawing a reciprocal graph through a point where its denominator is zero.
- Using points on the curve rather than points on the tangent to estimate instantaneous gradient.
- Omitting asymptotes or turning points from a sketch.
Knowledge Check And Practice
1. State the roots of a graph that crosses the x-axis at x=-3,1,4.
2. State the horizontal asymptote of y=2^x-5.
3. State the vertical asymptote of y=7/(x+3).
4. Is y=4(1.2)^x growth or decay?
5. How do you solve f(x)=6 from the graph y=f(x)?
6. How do you solve f(x)=g(x) graphically?
7. What features should a quadratic sketch show?
8. Why should two points far apart be used on a tangent?
