Worked example · Easy
Complete a table
Find \(y\) when \(x=-1,0,1,2\) for \(y=x^2-2x-3\).
Values are \(0,-3,-4,-3\). These produce points \((-1,0),(0,-3),(1,-4),(2,-3)\).
Chapter 28
Graphs of functions
Draw and interpret important function graphs, use graphs to solve equations, recognise growth and decay, and estimate the gradient of a curve using a tangent.
By the end of this chapter, you should be able to
- construct tables of values and draw smooth curves;
- recognise quadratic, cubic, reciprocal, root and exponential graph shapes;
- use intersections to solve equations graphically;
- interpret exponential growth and decay;
- estimate the gradient of a curve by drawing a tangent.
1
Subtopic 1
Quadratic graphs
Method, model and exam practice
Make a table of values on both sides of the turning point, plot accurately and join with a smooth curve. The roots are the x-coordinates where the graph crosses or touches the x-axis.
Worked example — guided method
For \(y=x^2-4x+3\), find the roots and turning point.
\(y=(x-1)(x-3)\), so roots are \(x=1,3\). Completing the square gives \(y=(x-2)^2-1\), so vertex \((2,-1)\).
Practice: basic to exam-style
1. Build confidence: Find the y-intercept of \(y=x^2+2x-8\).
Show Answer
Put \(x=0\): \(y=-8\). Intercept \((0,-8)\).
2. Exam-style application: For \(y=x^2-6x+8\), find the two x-intercepts.
Show Answer
\((x-2)(x-4)=0\), so intercepts are \((2,0)\), \((4,0)\).
A quadratic function has the form \(y=ax^2+bx+c\). Its graph is a smooth curve called a parabola. A positive \(a\) gives a U-shaped curve; a negative \(a\) turns it upside down.
Drawing a quadratic graph
- Choose the required x-values.
- Calculate y accurately in a table.
- Plot the coordinate pairs.
- Join the points with one smooth curve.
- Label the curve.
Worked example · Medium
Read roots from a graph
The graph of \(y=x^2-2x-3\) crosses the x-axis at \(x=-1\) and \(x=3\). Solve \(x^2-2x-3=0\).
At the x-axis, \(y=0\), so the roots are \(\boxed{x=-1,\ 3}\).
2
Subtopic 2
Using graphs to solve equations
Method, model and exam practice
To solve \(f(x)=k\), draw the horizontal line \(y=k\) and read the x-values at intersections. More general equations may need rearranging into a form visible on the plotted graph.
Worked example — guided method
Using a graph of \(y=x^2-2x-3\), solve \(x^2-2x-3=5\).
Draw \(y=5\). Algebraic check: \(x^2-2x-8=0=(x-4)(x+2)\), giving approximate graphical solutions \(x=-2,4\).
Practice: basic to exam-style
1. Build confidence: A graph of \(y=x^2\) is shown. How would you solve \(x^2=9\)?
Show Answer
Draw \(y=9\) and read two intersections, \(x=-3,3\).
2. Exam-style application: How do you solve \(x^2+3x=4\) on \(y=x^2+3x\)?
Show Answer
Draw horizontal \(y=4\); check \(x^2+3x-4=(x+4)(x-1)\), roots \(-4,1\).
Graphs solve equations through intersections. To solve \(f(x)=0\), read the x-values where the curve crosses the x-axis. To solve \(f(x)=k\), draw the horizontal line \(y=k\). For an equation such as \(f(x)=g(x)\), draw both graphs and read their intersection x-values.
Worked example · Medium
Solve \(f(x)=5\)
For \(f(x)=x^2-2x-8\), the graph meets \(y=5\) at approximately \(x=-2.7\) and \(x=4.7\).
So \(x^2-2x-8=5\) has approximate solutions \(\boxed{-2.7,\ 4.7}\).
Worked example · Exam-style
Rearrange before using the graph
You already have the graph of \(y=2x^2-x-3\). Explain how to solve \(2x^2-x=x+5\).
Subtract 3 from both sides:
\[2x^2-x-3=x+2\]Draw the line \(y=x+2\). Read the x-coordinates where it intersects the existing curve.
3
Subtopic 3
Cubic graphs
Method, model and exam practice
Cubic curves may have one or three real x-intercepts. The sign of the leading coefficient determines end behaviour. Make a careful table and plot enough points to capture turning points.
Worked example — guided method
For \(y=x^3-4x\), find the x-intercepts.
\(x^3-4x=x(x^2-4)=x(x-2)(x+2)\). Intercepts at \(-2,0,2\).
Practice: basic to exam-style
1. Build confidence: Find \(y\) at \(x=-2\) for \(y=x^3+2x\).
Show Answer
\((-2)^3+2(-2)=-8-4=-12\).
2. Exam-style application: What is the y-intercept of \(y=x^3-3x+5\)?
Show Answer
When \(x=0\), \(y=5\); intercept is \((0,5)\).
Cubic functions contain an \(x^3\) term. A basic positive cubic such as \(y=x^3\) runs from bottom-left to top-right and has a characteristic double bend.
Worked example · Easy
Use a cubic graph
The line \(y=12\) meets \(y=x^3\) at about \(x=2.3\).
Therefore \(x^3=12\) has graphical solution \(\boxed{x\approx2.3}\).
Worked example · Exam-style
Multiple roots
A graph of \(y=x^3-2x\) crosses the x-axis three times. What does that tell you?
The equation \(x^3-2x=0\) has three real roots. From the textbook graph they are approximately \(\boxed{-1.4,0,1.4}\).
4
Subtopic 4
Reciprocal graphs
Method, model and exam practice
A reciprocal graph \(y=k/x\) cannot have \(x=0\); its branches approach the axes but never cross them when \(k e0\). Use values of both signs when plotting.
Worked example — guided method
For \(y=12/x\), find points when \(x=-3,2,6\).
\(x=-3\Rightarrow y=-4\); \(x=2\Rightarrow y=6\); \(x=6\Rightarrow y=2\).
Practice: basic to exam-style
1. Build confidence: For \(y=-8/x\), find \(y\) when \(x=4\).
Show Answer
\(y=-8/4=-2\).
2. Exam-style application: Solve \(12/x=3\).
Show Answer
Multiply by \(x\ne0\): \(12=3x\), so \(x=4\).
Reciprocal functions include \(y=\frac{a}{x}\) and \(y=\frac{a}{x^2}\). Since division by zero is undefined, \(x=0\) is excluded. The curves approach the axes but do not meet them.
Shape clues
For \(y=\frac{a}{x}\), the two branches lie in opposite quadrants if \(a>0\). For \(y=\frac{a}{x^2}\) with \(a>0\), both branches lie above the x-axis.
Worked example · Medium
Why no point at x = 0?
Explain why \(y=\frac4x\) cannot have a point where \(x=0\).
It would require \(4\div0\), which is undefined. Therefore \(x=0\) is not in the domain.
5
Subtopic 5
Other graphs
Method, model and exam practice
Recognise the shape from the equation and the domain. Square-root functions require a non-negative radicand for real values; compare graphs of \(\sqrt x\) and \(\sqrt{x-a}\).
Worked example — guided method
For \(y=2\sqrt{x}\), calculate \(y\) at \(x=0,4,9\).
Values are \(0,2(2)=4,2(3)=6\), giving points \((0,0),(4,4),(9,6)\).
Practice: basic to exam-style
1. Build confidence: Find the smallest possible x for \(y=\sqrt{x-5}\).
Show Answer
The value inside the square root must be nonnegative: \(x-5\ge0\), hence \(x\ge5\).
2. Exam-style application: For \(y=|x-2|\), find \(y\) at \(x=-1\).
Show Answer
\(y=|-1-2|=|-3|=3\).
The chapter also includes root functions and functions that combine different powers of x. A square-root graph is restricted to values of x for which the expression under the root is non-negative.
Worked example · Easy
Domain restriction
Why is \(x=-4\) not used on the real graph of \(y=2\sqrt{x}\)?
\(\sqrt{-4}\) is not a real number, so the real graph only uses \(\boxed{x\ge0}\).
Worked example · Exam-style
Mixed function behaviour
Why can a function containing \(1/x\) change very rapidly close to \(x=0\)?
As \(x\) gets close to zero, \(|1/x|\) becomes very large, so the reciprocal term dominates the shape.
6
Subtopic 6
Exponential graphs
Method, model and exam practice
For \(y=a^x\), the y-intercept is 1. With \(a>1\) the curve increases, and when \(0
Worked example — guided method
For \(y=3^x\), evaluate at \(x=-1,0,2\).
\(3^{-1}=1/3\), \(3^0=1\), \(3^2=9\).
Practice: basic to exam-style
1. Build confidence: Solve \(2^x=32\).
Show Answer
\(32=2^5\), so \(x=5\).
2. Exam-style application: Is \(y=(0.6)^x\) increasing or decreasing?
Show Answer
Because the base 0.6 lies between 0 and 1, it is decreasing.
An exponential graph has a variable in the exponent, for example \(y=ka^x\). For \(a>1\), the values grow increasingly quickly as x increases.
Worked example · Medium
Read an exponential graph
From the textbook graph of \(y=3^x\), estimate y when \(x=2.4\).
The graph gives approximately \(\boxed{14}\).
Worked example · Medium
Solve an exponential equation graphically
Use the same graph to solve \(3^x=20\).
Draw \(y=20\) and read the x-value at the intersection: \(\boxed{x\approx2.7}\).
7
Subtopic 7
Growth and decay
Method, model and exam practice
Translate a percentage change into a multiplier before drawing or interpreting a graph: growth \(1+r\), decay \(1-r\), where \(r\) is the decimal rate. Repeated changes use powers.
Worked example — guided method
A population of 4000 grows 5% each year. Estimate after 3 years.
\(P=4000(1.05)^3=4630.5\), approximately 4631 people.
Practice: basic to exam-style
1. Build confidence: A machine worth $2400 depreciates 20% per year. Find value after 2 years.
Show Answer
\(2400(0.8)^2=1536\) dollars.
2. Exam-style application: A graph follows \(y=500(1.1)^t\). What is the initial value?
Show Answer
At \(t=0\), \(y=500(1.1)^0=500\).
Functions of the form \(P=A\times b^x\) model repeated percentage change. If \(b>1\), the graph shows growth. If \(0 What does 1.05 mean in \(P=700(1.05)^x\)? \(1.05=105\%\), so the quantity increases by \(\boxed{5\%}\) each period. The population model above is graphed. Estimate the population after 5.5 years. Reading at \(x=5.5\) gives about \(\boxed{920}\).
Worked example · MediumInterpret a multiplier
Worked example · Exam-styleEstimate from a growth graph
8
Subtopic 8
Estimating the gradient to a curve
Method, model and exam practice
At a specific point on a curve draw a tangent, not a chord; choose two points far apart on the tangent and compute the rise over run. State negative gradients with the correct sign.
Worked example — guided method
A tangent passes approximately through \((1,3)\) and \((5,15)\). Estimate the curve gradient.
Gradient \(=(15-3)/(5-1)=12/4=3\).
Practice: basic to exam-style
1. Build confidence: A tangent passes through \((2,11)\) and \((7,1)\). Calculate its gradient.
Show Answer
\((1-11)/(7-2)=-10/5=-2\).
2. Exam-style application: Estimate the gradient of \(y=x^2\) at \(x=3\) from a tangent.
Show Answer
The tangent has gradient approximately 6; a graphical estimate close to 6 is expected.
A curve does not have one constant gradient. To estimate its gradient at a particular point, draw a tangent that just touches the curve there, then calculate the tangent's gradient.
Tangent method
- Draw the tangent carefully at the required point.
- Choose two convenient points far apart on the tangent.
- Calculate \(\frac{\Delta y}{\Delta x}\).
- State that this is an estimate.
Worked example · Exam-style
Estimate the gradient
A tangent to \(y=x^2\) at \((2,4)\) passes approximately through \((1,0)\) and \((3,8)\). Estimate the gradient.
\[m=\frac{8-0}{3-1}=\boxed{4}\]
Chapter 28 practice
Attempt each question before opening the answer. For graph-drawing questions, the answer describes the points and features your graph should show.
A. Quadratic and graphical solutions
1Table of values[3 marks]
For \(y=x^2-4\), find y when \(x=-2,-1,0,1,2\).
Show Answer
\(0,-3,-4,-3,0\).
2Roots from a graph[2 marks]
A quadratic graph crosses the x-axis at \(-3\) and \(5\). Solve \(f(x)=0\).
Show Answer
\(\boxed{x=-3\text{ or }x=5}\).
3Horizontal-line method[2 marks]
You have drawn \(y=x^2+x-2\). What line should you draw to solve \(x^2+x-2=4\)?
Show Answer
Draw \(\boxed{y=4}\).
4Two-graph method[3 marks]
You have drawn \(y=x^2-3x\). What line can be added to solve \(x^2-3x=x+2\)?
Show Answer
Draw \(\boxed{y=x+2}\). The intersection x-values solve the equation.
B. Cubic, reciprocal and other graphs
5Cubic shape[2 marks]
Describe the overall direction of \(y=x^3\).
Show Answer
It runs from bottom-left to top-right with a double bend.
6Reciprocal restriction[2 marks]
Why is \(x=0\) excluded from \(y=7/x\)?
Show Answer
Division by zero is undefined.
7Inverse-square graph[2 marks]
For \(y=2/x^2\), is y positive or negative when \(x=-3\)?
Show Answer
\[y=\frac{2}{9}>0\]
So it is positive.
8Square-root domain[2 marks]
State the real domain of \(y=3\sqrt{x}\).
Show Answer
\(\boxed{x\ge0}\).
C. Exponential growth and decay
9Growth or decay?[2 marks]
Does \(y=500(1.08)^x\) model growth or decay?
Show Answer
Growth, because the multiplier is greater than 1.
10Percentage change[2 marks]
What percentage change is represented by multiplier 0.84?
Show Answer
\(0.84=84\%\), so it is a \(\boxed{16\%\text{ decrease}}\).
11Initial value[2 marks]
For \(P=1200(1.03)^x\), what is P when \(x=0\)?
Show Answer
\[P=1200(1)=\boxed{1200}\]
12Graph estimate idea[2 marks]
To solve \(2^x=15\) from a graph of \(y=2^x\), what line should you draw?
Show Answer
Draw \(\boxed{y=15}\).
D. Tangents and exam-style reasoning
13Tangent gradient[3 marks]
A tangent passes through \((2,5)\) and \((8,17)\). Find its gradient.
Show Answer
\[m=\frac{17-5}{8-2}=\frac{12}{6}=\boxed{2}\]
14Meaning of tangent[2 marks]
Why is a tangent used instead of the whole curve to estimate gradient at one point?
Show Answer
The curve's gradient changes, while the tangent represents the local direction at that point.
15Graphical equation[4 marks]
You have drawn \(y=x^3-2x\). Explain how to solve \(x^3-2x=3\).
Show Answer
Draw the horizontal line \(y=3\), then read the x-coordinate(s) where it intersects the cubic curve.
16Combined functions[4 marks]
Why should extra x-values near zero often be plotted for functions containing \(1/x\)?
Show Answer
Because the reciprocal term changes very rapidly close to zero, so extra points are needed to show the curve's shape accurately.
17Quadratic graph accuracy[4 marks]
A table gives two equal lowest y-values at consecutive x-values. Why might an extra midpoint x-value be useful?
Show Answer
The true turning point may lie between them and have an even lower y-value. The midpoint helps locate the turning point more accurately.
18Growth-model interpretation[4 marks]
A car's value is modelled by \(V=5000(0.85)^t\). State the initial value and annual percentage decrease.
Show Answer
Initial value: \(\boxed{5000}\). Multiplier 0.85 means the car retains 85%, so it decreases by \(\boxed{15\%}\) each year.
Chapter 28 high-grade checklist
Tables and plotting
I calculate values accurately and use enough points to show the correct curve shape.
Graphical solutions
I know which extra line or curve to draw and read x-values at intersections.
Graph families
I recognise quadratic, cubic, reciprocal, root and exponential shapes.
Restrictions
I remember that reciprocal graphs exclude x = 0 and square-root graphs may restrict x.
Growth and decay
I interpret exponential multipliers correctly.
Tangents
I estimate the gradient of a curve by drawing a good tangent and using two well-spaced points.