In pre-calculus, a function is a mathematical relationship between two sets of numbers. It assigns each input exactly one output. The graph of a function represents this relationship visually, usually with a curve or a straight line.

Key Points:

• Function: A function takes an input (often represented by x) and produces a corresponding output (often represented by f(x) or y).
• Graph of a function: The graph is a visual representation of the function’s rule. It shows how the output (y or f(x)) changes as the input (x) changes.
• The graph of a function can be a curve (for non-linear functions) or a straight line (for linear functions).

Types of Functions:

• Linear functions: A function is linear if its graph is a straight line. The equation for a linear function is usually written as:
  f(x) = mx + b
  where m is the slope (how steep the line is) and b is the y-intercept (the point where the line crosses the y-axis).
• Quadratic functions: A function is quadratic if its graph is a parabola (a U-shaped curve). The equation for a quadratic function is usually written as:
  f(x) = ax² + bx + c
  where a, b, and c are constants.

Example 1: Linear Function

Consider the linear function: f(x) = 2x + 3. This equation represents a straight line. To graph it, you need to plot points by substituting values for x and calculating the corresponding y values.
Let's substitute a few values for x:

• If x = 0, then f(0) = 2(0) + 3 = 3. So the point is (0, 3).
• If x = 1, then f(1) = 2(1) + 3 = 5. So the point is (1, 5).
• If x = -1, then f(-1) = 2(-1) + 3 = 1. So the point is (-1, 1).

You can plot these points on a graph, and you will see a straight line.

Example 2: Quadratic Function

Consider the quadratic function: f(x) = x² - 4x + 3. The graph of this function is a parabola. To graph it, we calculate the values of y for different values of x:

• If x = 0, then f(0) = (0)² - 4(0) + 3 = 3. So the point is (0, 3).
• If x = 1, then f(1) = (1)² - 4(1) + 3 = 0. So the point is (1, 0).
• If x = 2, then f(2) = (2)² - 4(2) + 3 = -1. So the point is (2, -1).

The graph of this function will be a U-shaped curve, opening upwards, with the vertex at (2, -1).

Summary:

Functions are mathematical relationships that take an input and produce an output. Their graphs represent how the output changes as the input changes. Linear functions produce straight-line graphs, while quadratic functions produce parabolic (U-shaped) graphs. Understanding the behavior of different functions and how to graph them is a key concept in pre-calculus.

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