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Graphing functions

MAT1510 - Precalculus Mathematics A · Functions and Graphs

Graphing Functions

Graphing functions is an essential skill in precalculus mathematics. It allows you to visualise how a function behaves over a range of input values. This topic will cover the process of graphing different types of functions, including linear, quadratic, polynomial, and trigonometric functions.

Understanding the Cartesian Plane

The Cartesian plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin, denoted as (0, 0). Each point on the plane can be described by an ordered pair (x, y), where x is the horizontal distance from the origin, and y is the vertical distance.

Linear Functions

A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept. The slope indicates how steep the line is, while the y-intercept is the point where the line crosses the y-axis.

Example: Graphing a Linear Function

Consider the function f(x) = 2x + 3.

  1. Identify the slope (m) and y-intercept (b):
m = 2, b = 3
  1. Plot the y-intercept (0, 3) on the graph.
  2. Use the slope to find another point. Since the slope is 2, you can rise 2 units and run 1 unit to the right from (0, 3). This gives you the point (1, 5).
  3. Draw a straight line through the points (0, 3) and (1, 5).

Remember: The slope can be expressed as a fraction. A slope of 2 means you can think of it as 2/1, meaning rise 2 and run 1.

Quadratic Functions

A quadratic function has the form f(x) = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola. If a is positive, the parabola opens upwards; if a is negative, it opens downwards.

Example: Graphing a Quadratic Function

Consider the function f(x) = x^2 - 4.

  1. Identify the coefficients: a = 1, b = 0, c = -4.
  2. Find the vertex using the formula x = -b/(2a):
x = -0/(2*1) = 0
  1. Substitute x = 0 into the function to find the y-coordinate of the vertex:
f(0) = 0^2 - 4 = -4
  1. The vertex is at (0, -4).
  2. Find the x-intercepts by setting f(x) = 0:
x^2 - 4 = 0
(x - 2)(x + 2) = 0
  1. This gives x = 2 and x = -2 as the x-intercepts. Plot the points (2, 0) and (-2, 0).
  2. Draw the parabola through the vertex and the x-intercepts.

Watch out: Be careful with the direction of the parabola. If a is negative, the parabola opens downwards.

Polynomial Functions

A polynomial function can have multiple terms and is expressed as f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0, where n is a non-negative integer and a_i are constants. The degree of the polynomial is the highest power of x.

Example: Graphing a Polynomial Function

Consider the function f(x) = x^3 - 3x^2 - 4x + 12.

  1. Identify the degree (n = 3) and leading coefficient (a = 1).
  2. Find the x-intercepts using synthetic division or factoring. For simplicity, we will use trial and error to find one root:
f(2) = 2^3 - 3(2^2) - 4(2) + 12 = 0
  1. Since x = 2 is a root, we can divide the polynomial by (x - 2) using synthetic division:
2 | 1  -3  -4  12
  |    2   -2  -12
------------------
    1  -1  -6   0
  1. The quotient is x^2 - x - 6, which can be factored as (x - 3)(x + 2).
  2. Thus, the x-intercepts are x = 2, x = 3, and x = -2.
  3. Plot the points (2, 0), (3, 0), and (-2, 0).
  4. Determine the behaviour of the graph as x approaches positive and negative infinity. Since the leading coefficient is positive and the degree is odd, the graph will rise to positive infinity on the right and fall to negative infinity on the left.
  5. Draw the curve through the points, ensuring it reflects the end behaviour.

Remember: The number of x-intercepts can be less than or equal to the degree of the polynomial.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, have periodic behaviour, meaning they repeat their values in regular intervals. The sine function is defined as f(x) = sin(x) and the cosine function as f(x) = cos(x).

Example: Graphing a Sine Function

Consider the function f(x) = sin(x).

  1. The sine function has a period of 2π, meaning it repeats every 2π units.
  2. The amplitude is 1, which means the maximum and minimum values are 1 and -1, respectively.
  3. Plot key points:
f(0) = sin(0) = 0
f(π/2) = sin(π/2) = 1
f(π) = sin(π) = 0
f(3π/2) = sin(3π/2) = -1
f(2π) = sin(2π) = 0
  1. Draw the curve, ensuring it smoothly oscillates between -1 and 1.

Tip: Use a graphing calculator to check your work when graphing trigonometric functions.

Summary

  • Graphing functions involves plotting points on the Cartesian plane.
  • Linear functions have a constant slope and form straight lines.
  • Quadratic functions form parabolas and can be identified by their vertex and intercepts.
  • Polynomial functions can have multiple terms and varying degrees, affecting their shape.
  • Trigonometric functions are periodic and oscillate between specific values.

Check your understanding

  1. What is the general form of a linear function?
  2. How do you find the vertex of a quadratic function?
  3. What is the significance of the leading coefficient in a polynomial function?
  4. Describe the key characteristics of the sine function.