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Chapter 1: Problem 26

Graph each function "by hand." [Note: Even if you have a graphing calculator, it is important to be able to sketch simple curves by finding a few important points.] $$ f(x)=-3 x+5 $$

Short Answer

Expert verified
Plot the y-intercept at (0, 5), use slope -3, and draw the line through points (0, 5) and (1, 2).

Step by step solution

01

Identify the Type of Function

The given function is a linear function because it can be written in the form of \(f(x) = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, \(m = -3\) and \(c = 5\).
02

Determine the Y-Intercept

To determine the y-intercept, set \(x = 0\) in the function. Calculate \(f(0) = -3(0) + 5 = 5\). Thus, the y-intercept is \(5\), meaning the graph will cross the y-axis at the point \((0, 5)\).
03

Determine Another Point Using the Slope

The slope \(m = -3\) indicates that for every unit increase in \(x\), \(f(x)\) decreases by 3 units. Starting from point \((0, 5)\), choose \(x = 1\). Calculate \(f(1) = -3(1) + 5 = 2\). So, another point on the graph is \((1, 2)\).
04

Plot the Points on a Graph

On a Cartesian plane, plot the points \((0, 5)\) and \((1, 2)\). These points represent the intercept and an additional point along the line of the function.
05

Draw the Line

Use a ruler to draw a straight line through the points \((0, 5)\) and \((1, 2)\). This line represents the graph of the function \(f(x) = -3x + 5\). Extend the line in both directions to cover the graph.

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Key Concepts

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

Graphing
Graphing a linear function involves plotting points on a Cartesian coordinate plane and connecting them with a straight line.
To graph the function, start by identifying key points such as the y-intercept and additional points derived from the slope.
This process visually represents the equation on a two-dimensional plane, providing a clear picture of the function's behavior.
  • Begin at the y-intercept, the point where the curve crosses the y-axis.
  • Locate additional points using the slope to ensure accuracy.
  • Draw a straight line through these points, extending the line across the plane.
By sketching the line manually, you gain a better understanding of its direction and intersecting points, crucial for solving and predicting outcomes in algebraic contexts.
Slope
The slope of a linear function measures how steep the line is. It tells you how much the function value changes for one unit increase in the x-variable.
The slope, represented by the variable \(m\), is a key component in the slope-intercept form of a linear equation, \(f(x) = mx + c\).
In the given function, \(f(x) = -3x + 5\), the slope is \(-3\).
  • A negative slope indicates that the line slopes downward as you move from left to right.
  • The slope of \(-3\) means for every increase of 1 in \(x\), \(f(x)\) decreases by 3 units.
Understanding slope is essential for predicting the movement of the line and evaluating the rate of change. In real-world contexts, slope can represent mortality trends, economic inflation, or any scenario where changes occur at a consistent rate.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis. It is a vital component of a linear function, as it indicates where the line will intersect the vertical axis when \(x = 0\).
In the function \(f(x) = -3x + 5\), the y-intercept is the constant \(c\), which is \(5\).
  • Locate the y-intercept by setting \(x = 0\) in the equation.
  • Calculate the resulting \(f(x)\) to find the intercept point on the y-axis.
  • This point helps define the initial position of the line before slope adjustments are applied.
Recognizing the y-intercept allows you to establish the starting point for graphing the line and provides a foundational point from which to plot additional points using the slope.

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Most popular questions from this chapter

ATHLETICS: Juggling If you toss a ball \(h\) feet straight up, it will return to your hand after \(T(h)=0.5 \sqrt{h}\) seconds. This leads to the juggler's dilemma: Juggling more balls means tossing them higher. However, the square root in the above formula means that tossing them twice as high does not gain twice as much time, but only \(\sqrt{2} \approx 1.4\) times as much time. Because of this, there is a limit to the number of balls that a person can juggle, which seems to be about ten. Use this formula to find: a. How long will a ball spend in the air if it is tossed to a height of 4 feet? 8 feet? b. How high must it be tossed to spend 2 seconds in the air? 3 seconds in the air?

For any \(x\), the function \(\operatorname{INT}(x)\) is defined as the greatest integer less than or equal to \(x\). For example, \(\operatorname{INT}(3.7)=3\) and \(\operatorname{INT}(-4.2)=-5\) a. Use a graphing calculator to graph the function \(y_{1}=\operatorname{INT}(x) .\) (You may need to graph it in DOT mode to eliminate false connecting lines.)

BUSINESS: Isocost Lines An isocost line (iso means "same") shows the different combinations of labor and capital (the value of factory buildings, machinery, and so on) a company may buy for the same total cost. An isocost line has equation $$ w L+r K=\mathrm{C} \quad \text { for } L \geq 0, \quad K \geq 0 $$ where \(L\) is the units of labor costing \(w\) dollars per unit, \(K\) is the units of capital purchased at \(r\) dollars per unit, and \(C\) is the total cost. Since both \(L\) and \(K\) must be nonnegative, an isocost line is a line segment in just the first quadrant. a. Write the equation of the isocost line with \(w=8, \quad r=6, \quad C=15,000\), and graph it in the first quadrant. b. Verify that the following \((L, K)\) pairs all have the same total cost. \((1875,0),(1200,900),(600,1700),(0,2500)\)

$$ \begin{array}{l} \text { True or False: If } f(x)=m x+b, \text { then }\\\ f(x+h)=f(x)+m h \end{array} $$

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