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Chapter 0: Problem 16

Graph. List the slope and \(y\) -intercept. $$g(x)=x-2.5$$

Short Answer

Expert verified
Slope: 1, y-intercept: -2.5

Step by step solution

01

Identify the Slope-Intercept Form

Recognize that the given equation is in the slope-intercept form of a linear equation, which is given by: y = mx + bwhere m = slopeb = y-intercept.
02

Determine the Slope

Identify the coefficient of the variable x in the equation. The given equation is:g(x) = x - 2.5Here, the coefficient of x (m) is 1. Hence, the slope (m) is 1.
03

Determine the y-intercept

Identify the constant term in the equation. The constant term in the given equation g(x) = x - 2.5 is -2.5. Therefore, the y-intercept (b) is -2.5.
04

State the Slope and y-intercept

The slope of the graph is 1, and the y-intercept is -2.5. Therefore, the graph of the function g(x) = x - 2.5 has a slope of 1 and a y-intercept of -2.5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are mathematical statements that describe a straight line when graphed on a coordinate plane.
They follow the format: y = mx + bwhere y is the output or dependent variable, x is the input or independent variable, m is the slope, and b is the y-intercept.

For example, in the given equation g(x)=x-2.5, we can see it fits this format perfectly with m (slope) being 1 and b (y-intercept) being -2.5.
Understanding linear equations is key because they are foundational to more complex topics in algebra and calculus.
Slope
The slope of a line is a measure of its steepness and direction.
Mathematically, it is often represented by the letter m.
In the slope-intercept form of a linear equation, y = mx + b, the slope tells us how much the y value changes for a given change in the x value.
For example, a slope of 1 means that for every increase of 1 unit in x, y also increases by 1 unit.

In our given equation g(x)=x-2.5, the slope m is 1. This indicates a directly proportional and linear relationship between x and y.
y-intercept
The y-intercept is the point where the line crosses the y-axis.
In the slope-intercept form equation y = mx + b, b is the y-intercept. This value shows the y value when x is zero.

For the equation g(x)=x-2.5, our y-intercept (b) is -2.5.
This means that when x is 0, the output is -2.5 and the line passes through the point (0, -2.5) on the graph.
Graphing Functions
Graphing linear functions helps visualize the relationship between variables and makes it easier to interpret results.
To graph g(x)=x-2.5, you would start by plotting the y-intercept (0, -2.5).
Next, use the slope to find another point on the line. Since the slope is 1, from the y-intercept go up 1 unit and right 1 unit to plot the next point (1, -1.5).

Connect these points with a straight line and extend it in both directions to complete the graph.

This visual representation can greatly assist in understanding how changes in x affect y.

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