Problem 32 - Algebra A Combined Function

Question

Sketch the graph of each function. $$ f(x)=(x-1)+4 $$

Step-by-Step Solution

Answer

This linear function has a y-intercept at (0, 3) and a slope of 1. Plot (0, 3), (1, 4), and (-1, 2), then draw a line through them.

Step 1: Identify Function Type

The given function is \( f(x) = (x - 1) + 4 \). This is a linear function as it is of the form \( f(x) = mx + b \), where \( m = 1 \) and \( b = 3 \), after simplifying it to \( f(x) = x + 3 \).

Step 2: Find the Y-Intercept

The y-intercept of a linear function \( y = mx + b \) is the point where the graph crosses the y-axis. It occurs when \( x = 0 \). Substitute \( x = 0 \) into the equation:\[ f(0) = 0 + 3 = 3 \]So, the y-intercept is (0, 3).

Step 3: Find the Slope

The slope \( m \) of the function \( f(x) = x + 3 \) is the coefficient of \( x \), which equals 1. This means that for every unit increase in \( x \), \( f(x) \) increases by 1.

Step 4: Plot Additional Points

Using the y-intercept and the slope, let's find another point. Starting at (0, 3), increase \( x \) by 1:- When \( x = 1 \), \( f(x) = 1 + 3 = 4 \). So, another point is (1, 4).- Another point: When \( x = -1 \), \( f(x) = -1 + 3 = 2 \). So, another point is (-1, 2).

Step 5: Sketch the Graph

Plot the points (0, 3), (1, 4), and (-1, 2) on a graph. Draw a straight line through these points since a linear function forms a straight line. The graph extends infinitely in both directions along this line.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves drawing a line on a coordinate plane based on an equation in the form \( y = mx + b \). In the case of the equation \( f(x) = x + 3 \), graphing requires identifying crucial elements like the slope \( m \) and the y-intercept \( b \) to determine the line's position and direction on the graph.
A linear graph is characterized by its straight-line appearance, indicating a constant rate of change. The process starts by marking the y-intercept on the y-axis and using the slope to determine two or more points through which the line passes. This method ensures accurate representation for students who need a visual understanding of how linear functions behave.
Understanding how to graph these equations is foundational in learning more complex concepts like systems of equations and inequalities.

Y-Intercept

The y-intercept is a fundamental concept in graphing linear equations, denoting where the graph crosses the y-axis. For our function \( f(x) = x + 3 \), the y-intercept is 3, meaning the line passes through the point (0, 3) on the graph.
Calculating the y-intercept involves setting \( x = 0 \) and solving the equation, which simplifies to the constant term \( b \) in \( y = mx + b \).
Key points about the y-intercept:

• The y-intercept is always the point where the value of \( x \) is zero.
• It provides a starting point for plotting the line on the graph.
• Understanding this intersection helps in predicting the function's behavior as part of a larger graph.

Identifying the y-intercept is crucial as it anchors the graph, allowing one to apply the slope to determine the rest of the points.

Slope of a Line

The slope of a line in a linear equation measures its steepness and direction. For our function \( f(x) = x + 3 \), the slope \( m \) is 1, indicating that for each increase by 1 in \( x \), \( f(x) \) increases by 1 as well.
To calculate the slope, consider the formula \( m = \frac{\Delta y}{\Delta x} \), which is the change in the y-value over the change in the x-value. A positive slope means the line rises to the right, while a negative slope would indicate it falls.
Key considerations about slopes:

• The slope dictates the angle of the line on the graph.
• It is a constant for any linear function, ensuring the uniform direction of the line.
• Analyzing slopes helps understanding whether a function is increasing or decreasing.

The slope connects every point on a linear function, creating the consistent and predictable straight line characteristic of linear graphs.

Plotting Points

Plotting points is a precise method to sketch the graph of a function. In this case, for \( f(x) = x + 3 \), it's about establishing points that the line will travel through.
Begin by identifying key points, starting with the y-intercept. From the y-intercept, use the slope to find other points. For example, starting from (0, 3) with a slope of 1:

• Increasing \( x \) by 1 to get \( f(1) = 4 \), adding point (1, 4).
• Decreasing \( x \) by 1 to get \( f(-1) = 2 \), adding point (-1, 2).

This method of plotting multiple points and drawing a line through them accurately reflects the graph of the function.
It confirms the behavior of the equation and also helps verify calculations, ensuring the function's correct representation on the graph.