Graphs of Standard Functions
This topic covers the essential shapes of common mathematical functions. The ability to quickly sketch and interpret these graphs is vital for solving problems visually, such as finding the number of solutions to an equation by looking for intersections, without needing a calculator.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Linear: A straight line where `m` is the gradient (steepness) and `c` is the y-intercept.`y = mx + c`
- Quadratic: A parabola. If `a > 0`, it's U-shaped; if `a < 0`, it's n-shaped. Key features are its roots (x-intercepts) and turning point.`y = ax2 + bx + c`
- Simple Cubic: An 'S' shaped curve passing through the origin (0,0). Its orientation depends on the sign of `a`.`y = ax3`
- Reciprocal: Has two separate branches in opposite quadrants (1st and 3rd). It never touches the axes; the axes are its asymptotes.`y = 1/x`
- Exponential: Always passes through (0,1). For `k > 1`, it shows rapid growth. For `0 < k < 1`, it shows decay towards the x-axis.`y = k^x`
- Trigonometric (`sin x`, `cos x`, `tan x`): These are periodic functions. `sin x` and `cos x` oscillate between -1 and 1. `tan x` has a period of 180° and vertical asymptotes.
Diagram
Formulae
y = mx + c To identify the gradient (m) and y-intercept (c) of a straight line from its equation.
y = ax2 + bx + c To identify a quadratic function. The sign of 'a' determines if the parabola is U-shaped (a > 0) or n-shaped (a < 0).
Definitions
- Intercept
- The point where a graph crosses one of the coordinate axes. The y-intercept is where x=0, and the x-intercept is where y=0.
- Root
- A solution to an equation of the form f(x) = 0. On a graph, roots are the x-coordinates of the x-intercepts.
- Asymptote
- A line that a curve approaches increasingly closely but never touches or crosses.
- Turning Point
- A point on a curve where the gradient changes sign, such as the minimum or maximum point of a parabola.
Worked example
By sketching the graphs of y = -x2 and y = 1/x on the same axes, determine the number of real solutions to the equation -x2 = 1/x.
- 1
First, identify the shape of y = -x2.
The x2 term indicates a parabola, and the negative sign means it is an 'n-shaped' curve with its maximum at the origin (0,0).
- 2
Next, identify the shape of y = 1/x.
This is a reciprocal graph with two branches, one in the first quadrant (where x and y are positive) and one in the third quadrant (where x and y are negative).
- 3
Sketch both graphs on the same set of axes.
The n-shaped parabola is entirely in the 3rd and 4th quadrants (except for the origin).
The reciprocal graph is in the 1st and 3rd quadrants.
- 4
Look for points of intersection.
The parabola's branch in the 3rd quadrant will clearly intersect with the reciprocal graph's branch in the 3rd quadrant.
- 5
The parabola's branch in the 4th quadrant and the reciprocal's branch in the 1st quadrant will never meet.
- 6
Therefore, there is only one point of intersection.
Answer: 1
Common mistakes
- ×Mistaking the effect of a negative coefficient. For example, sketching y = -x3 as an increasing function instead of a decreasing one, which is a reflection of y = x3 in the x-axis.
- ×Incorrectly sketching exponential decay. The graph of y = (1/3)^x should decrease as x increases, not increase.
- ×Forgetting that the reciprocal function y = 1/x is undefined at x=0 and has branches in two separate quadrants.
No-calculator tips
- ✓To quickly sketch a graph, plot values for x = -1, 0, and 1. This helps anchor the curve's position and shape.
- ✓To compare two functions, like x2 and x3, test a value between 0 and 1, such as x=1/2. Here, (1/2)2 = 1/4 and (1/2)3 = 1/8, showing x2 is above x3 in this interval.
- ✓Many questions ask for the *number* of solutions, not the solutions themselves. A good sketch showing the number of intersections is often sufficient and much faster than solving algebraically.