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Magazine Chapter 3: Problem 87
Graph the linear function and state the domain and range. $$ C(x)=100+50 x $$
Short Answer
Expert verified
The domain and range of the function are both all real numbers.
Step by step solution
01
Identify the Type of Function
The function given is a linear function, expressed in the slope-intercept form as \(C(x) = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. For \(C(x) = 100 + 50x\), the slope \(m = 50\) and the y-intercept \(b = 100\).
02
Determine the Slope and Y-intercept
From the equation \(C(x) = 100 + 50x\), identify the slope \(m = 50\), which means the line rises 50 units vertically for every 1 unit horizontal increase. The y-intercept \(b = 100\) indicates that the line crosses the y-axis at the point \((0, 100)\).
03
Choose Points to Plot
To graph the line, choose at least two points. We already have one point \((0, 100)\) because of the y-intercept. For another point, you can calculate \(C(x)\) for a different value of \(x\), such as \(x = 1\). Then, \(C(1) = 100 + 50(1) = 150\), giving the point \((1, 150)\).
04
Plot the Points on the Graph
On a coordinate plane, plot the points \((0, 100)\) and \((1, 150)\). These points form a straight line representing the linear function.
05
Draw the Line
Using a ruler, draw a straight line through the points \((0, 100)\) and \((1, 150)\). Extend the line in both directions, ensuring it spans the entire graph window.
06
State the Domain
The domain of a linear function is all real numbers, so the domain is \(x \in (-\infty, \infty)\).
07
State the Range
Since the function is a line that extends indefinitely in both the positive and negative directions vertically, the range is also all real numbers, or \(C(x) \in (-\infty, \infty)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Slope-Intercept Form
The slope-intercept form is a standard way to express linear equations. It is typically written as \(y = mx + b\). In this equation:
The y-intercept, \(b\), is the point where the line crosses the y-axis, meaning \(x = 0\). For instance, in our given equation \(C(x) = 100 + 50x\), the slope \(m\) is 50, indicating a steep upwards incline. The y-intercept \(b\) is 100, which is where the line hits the y-axis.
- \(m\) represents the slope of the line
- \(b\) is the y-intercept.
The y-intercept, \(b\), is the point where the line crosses the y-axis, meaning \(x = 0\). For instance, in our given equation \(C(x) = 100 + 50x\), the slope \(m\) is 50, indicating a steep upwards incline. The y-intercept \(b\) is 100, which is where the line hits the y-axis.
Exploring Domain and Range
The domain and range are fundamental concepts when discussing functions. They describe the set of possible inputs and outputs respectively. For linear functions:
- The domain encompasses all potential x-values. Since a linear function like \(C(x) = 100 + 50x\) does not have any restrictions, the domain is all real numbers, expressed as \(x \in (-\infty, \infty)\).
- The range signifies all possible y-values. For linear functions that don鈥檛 have a maximum or minimum value, like ours, the range is also all real numbers, \(C(x) \in (-\infty, \infty)\).
Graphing Linear Equations
Graphing linear equations is simple once you understand the components involved. Start by identifying the slope and y-intercept, as detailed in the slope-intercept form section.
1. **Plot the y-intercept**: Begin by marking the point where the line crosses the y-axis. For example, for \(C(x) = 100 + 50x\), this point is \((0, 100)\).
2. **Use the slope to find another point**: From the y-intercept, apply the slope to determine another point on the line. With a slope of 50, for every 1 unit increase in \(x\), \(y\) increases by 50 units. If \(x = 1\), then \(y = 150\), marking your second point at \((1, 150)\).
3. **Draw the line**: With both points plotted, use a ruler to draw a straight line through them. Extend the line across your graph. This visual representation helps understand the behavior of linear functions.
1. **Plot the y-intercept**: Begin by marking the point where the line crosses the y-axis. For example, for \(C(x) = 100 + 50x\), this point is \((0, 100)\).
2. **Use the slope to find another point**: From the y-intercept, apply the slope to determine another point on the line. With a slope of 50, for every 1 unit increase in \(x\), \(y\) increases by 50 units. If \(x = 1\), then \(y = 150\), marking your second point at \((1, 150)\).
3. **Draw the line**: With both points plotted, use a ruler to draw a straight line through them. Extend the line across your graph. This visual representation helps understand the behavior of linear functions.
