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Chapter 4: Problem 49

Graph each function. $$C(h)=1.5 h$$

Short Answer

Expert verified
The linear function is given by \(C(h) = 1.5h\), with a slope of 1.5 and y-intercept of 0. To graph the function, first plot two points on the line, such as (0,0) and (2,3). Then, draw a straight line through these points to represent the graph of the function.

Step by step solution

01

1. Identify the slope and y-intercept

In the given function, $$C(h) = 1.5h,$$ we can see that the slope (m) is 1.5 and the y-intercept (b) is 0. This means that the function can be written as $$y = 1.5x$$ for graphing purposes.
02

2. Find two points on the line

Since the y-intercept is 0, we know that the line passes through the point (0,0). Next, we can find another point on the line by plugging in a value for h. Let's choose h = 2: $$C(2) = 1.5(2) = 3.$$ So, the line passes through the point (2,3).
03

3. Draw the line

Plot the points (0,0) and (2,3) on the coordinate plane. Then, draw a straight line through these points. This line represents the graph of the given function, $$C(h) = 1.5h.$$

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Key Concepts

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

Slope and Y-Intercept
Understanding the slope and y-intercept is crucial for graphing linear functions. In the function \(C(h) = 1.5h\), the slope is represented by the coefficient of \(h\), which is 1.5. The slope tells us how steep the line is and the rate at which \(C(h)\) changes with respect to \(h\). A positive slope indicates the line rises as you move from left to right, while a negative slope means it falls.
  • Slope (m): For our function, \(m = 1.5\).
  • Y-intercept (b): This is where the line crosses the y-axis, occurring when \(h = 0\). In our function, this is \(b = 0\).
This function, rewritten as \(y = 1.5x\), means the graph will pass through the origin (0,0) since the y-intercept is 0. Being familiar with the slope and y-intercept helps you predict the position and direction of a line even before graphing it.
Coordinate Plane
The coordinate plane is where we graph our linear equations. It is divided into four quadrants by two axes: the horizontal x-axis and the vertical y-axis. Each point on the plane is identified by an ordered pair \((x, y)\).
For graphing linear equations such as \(C(h) = 1.5h\), you need to plot at least two points to form a straight line, as a line is determined by two distinct points:
  • Point (0,0): With \(b = 0\), the line crosses the y-axis here.
  • Point (2,3): By substituting \(h = 2\) into the equation, \(C(2) = 3\), we get this point.
Plot these points accurately on the coordinate plane. Then, draw a line through them to represent the equation. Understanding how to use the coordinate plane is essential to accurately graphing and interpreting linear equations.
Linear Equations
Linear equations are fundamental in mathematics due to their simplicity and wide application. A linear equation is one in which the highest power of the variable is one, such as \(y = 1.5x\) for the equation \(C(h) = 1.5h\).
These equations often represent real-world relationships where one variable depends directly in proportion to another. For instance, this exercise exemplifies how the cost \(C\) changes with respect to \(h\) by a consistent amount (slope) without any initial cost \(y ext{-intercept} = 0\).
  • Linear relationships are expressed in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
  • The graph of a linear equation is always a straight line, which simplifies interpretation and prediction.
Recognizing these components and their roles will enhance your ability to solve problems involving linear equations and apply them effectively in various contexts.

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Most popular questions from this chapter

Write an equation of the line perpendicular to the given line and containing the given point. Write the answer in slope-intercept form or in standard form, as indicated. $$y=-5 x+1 ;(10,0) ; \text { standard form }$$

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=-\frac{3}{8 x-5}$$

Write the slope-intercept form of the equation of the line, if possible, given the following information. horizontal line containing \((0,-8)\)

Graph each function by finding the \(x\) - and \(y\) -intercepts and one other point. $$k(x)=-2 x+6$$

Write the slope-intercept form of the equation of the line, if possible, given the following information. contains \((3,0)\) and \((7,-2)\)
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