Question

Graph the functions (a) \(y=16+2 x\) (b) \(y=8-2 x\) (c) \(y=2 x+12\) (in each case, consider the domain as consisting of nonnegative real numbers only.)

Short answer

Graph each function in the first quadrant, ensuring lines only extend over nonnegative x-values.

Step-by-step solution

Understanding the Structure of Linear Equations

All three functions given are in the form of linear equations: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Our task is to graph these equations considering only nonnegative real numbers for the domain of \(x\).

Graphing y=16+2x

For the function \(y = 16 + 2x\), the slope \(m\) is 2, which means the line rises 2 units for each unit it moves to the right, and the y-intercept \(b\) is 16. Start by plotting the y-intercept (0, 16) on the graph, then use the slope to plot another point, for example, (1, 18), and draw a line through these points.

Graphing y=8-2x

The function \(y = 8 - 2x\) has a slope of -2 and a y-intercept of 8, indicating that for every unit increase in \(x\), \(y\) decreases by 2 units. Plot the point (0, 8) and use the slope to find another point, such as (1, 6), then draw a line through the points for this function.

Graphing y=2x+12

The function \(y = 2x + 12\) has a slope of 2 and a y-intercept of 12. Begin by plotting the y-intercept (0, 12). Using the slope, plot another point like (1, 14), and then draw the line connecting these points.

Analyzing the Domain

Since the domain is limited to nonnegative real numbers, ensure that each graph is only drawn in the first quadrant, starting from the y-intercept and extending to the right along the x-axis. Any part of the lines extending to negative \(x\) or \(y\) should not be included in the graph.

Key concepts

Graphing

When we hear about graphing functions, we are talking about creating a visual representation of mathematical equations on a coordinate plane. Let's break this down using the concept of linear equations, which are often represented in the form of a straight line.

• Start by identifying the components of the equation, usually given in the standard form like \(y = mx + b\).
• The coordinate plane features two axes: the horizontal axis (x-axis) and the vertical axis (y-axis).
• To plot the line represented by the equation, locate the y-intercept, which is the point where the line crosses the y-axis.
• Use the slope of the line to determine its steepness and direction.

In this context, graphing involves accurately plotting the line based on these indicators, ensuring that it reflects the equation's behavior accurately on the coordinate plane.

Slope-Intercept Form

The slope-intercept form of a linear equation is a convenient way to express the equation of a line. It is typically given as \(y = mx + b\) where:

• \(m\) is the slope of the line. The slope is a ratio that describes how steep the line is. It's calculated as the rise (change in \(y\)) over the run (change in \(x\)).
• A positive slope indicates the line rises left to right, while a negative slope means it falls.
• \(b\) is the y-intercept. This is the point at which the line crosses the y-axis, meaning it has the coordinates (0, \(b\)).

This form makes it easy to determine and plot both the slope and the y-intercept on a graph quickly. Knowing both these elements allows us to draw the line representing the linear equation efficiently and accurately.

Nonnegative Domain

In mathematics, a domain refers to the set of all possible input values (x-values) for a particular function. When considering a nonnegative domain, we are focusing solely on nonnegative real numbers—those equal or greater than zero.

• This criterion means the graph should only be plotted from the point where \(x = 0\) and extend to the right along the x-axis.
• For linear equations like those in our exercise, ensure that no part of the graph extends into areas where \(x\) would be negative.
• This often results in only showing the line within the first quadrant of the graph.

Working within a nonnegative domain is particularly relevant for scenarios where negative inputs aren't practical or required based on the context of a problem, such as certain real-world situations where only nonnegative values make sense.