Problem 8 - College Algebra with Modeling and Visualization

Question

Graph \(y=f(x)\) by hand by first plotting points to determine the shape of the graph. $$ f(x)=x+1 $$

Step-by-Step Solution

Answer

The graph of \( f(x) = x + 1 \) is a straight line with a slope of 1 and y-intercept at 1.

Step 1: Understand the Function

The given function is \( f(x) = x + 1 \), which represents a linear function. The graph of a linear function forms a straight line. The slope of this line is 1 and its y-intercept is 1.

Step 2: Create a Table of Values

To plot points, create a table of values by choosing different values for \( x \), then calculate the corresponding \( y \) values. For example, use \( x = -1, 0, 1, 2, 3 \).- \( f(-1) = -1 + 1 = 0 \)- \( f(0) = 0 + 1 = 1 \)- \( f(1) = 1 + 1 = 2 \)- \( f(2) = 2 + 1 = 3 \)- \( f(3) = 3 + 1 = 4 \)The points are \((-1, 0)\), \((0, 1)\), \((1, 2)\), \((2, 3)\), \((3, 4)\).

Step 3: Plot the Points

On a graph, plot the points \((-1, 0)\), \((0, 1)\), \((1, 2)\), \((2, 3)\), and \((3, 4)\). Use a Cartesian plane with both x and y axes labeled, and ensure that the increments are consistent for accurate representation.

Step 4: Draw the Line

Once the points are plotted, draw a straight line through all the points. Because this is a linear function, all points should align along a single straight line, indicating the graph of \( f(x) = x + 1 \).

Step 5: Verify the Line

Check that the line passes through more points not originally plotted to ensure accuracy. For example, for \( x = -2 \), \( f(-2) = -2 + 1 = -1 \), and ensure this point is located on the line. Confirm additional points to ensure the line's correct position.

Key Concepts

Linear Equations

A linear equation is a mathematical statement that describes a straight line when graphed on a coordinate plane. The general form of a linear equation is \( y = mx + b \), where:

• \( y \) is the dependent variable.
• \( m \) is the slope of the line, which indicates its steepness.
• \( x \) is the independent variable.
• \( b \) is the y-intercept, or the point where the line crosses the y-axis.

In the example \( f(x) = x + 1 \), the slope \( m \) is 1, and the y-intercept \( b \) is 1. This means as we move one unit along the x-axis, the y value increases by one unit, resulting in a perfectly diagonal line that rises to the right.

Plotting Points

Plotting points involves finding specific coordinates that satisfy the equation of a line and marking them on the coordinate plane, creating a visual representation of the equation. To do this effectively:

• Choose a few values for \( x \).
• Calculate the corresponding \( y \) values using the equation.
• Write down these pairs of \( (x, y) \) values as coordinates, like \((-1, 0)\), \((0, 1)\), etc.
• Plot these points on the graph paper.
• Ensure the points are accurately placed and in relative proportion.

This step bridges the mathematical equation and its graphical representation, helping identify the pattern or path the line follows.

Coordinate Plane

The coordinate plane is a two-dimensional surface with two axes, the x-axis, and the y-axis, intersecting at a point called the origin. This system allows us to describe the location of points in a flat space using ordered pairs. Here's how it works:

• The x-axis is horizontal and represents the input values of the function.
• The y-axis is vertical and shows the output values after applying the function.
• Coordinates are written as \((x, y)\), where \(x\) is the position on the horizontal axis, and \(y\) is the position on the vertical axis.
• This coordinate system is essential for plotting mathematical functions and visualizing their graphs.

Understanding the coordinate plane is crucial because it serves as the framework for graphing functions like linear equations, showing how changes in one variable correlate with changes in another.

Y-Intercept

The y-intercept of a linear equation is an important feature that provides valuable information about the graph's relation to the y-axis. It is the point where the line crosses the y-axis, giving us the value of \( y \) when \( x \) is zero.

• In the equation \( y = mx + b \), \( b \) represents the y-intercept.
• To find the y-intercept, simply evaluate the equation at \( x = 0 \).
• For example, in the function \( f(x) = x + 1 \), when \( x = 0 \), \( y = 0 + 1 = 1 \).
• This means the y-intercept is the point \((0, 1)\) on the graph.

Recognizing the y-intercept allows you to start graphing the line on the coordinate plane quickly. It acts as a reference point from which the line extends according to its slope.