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Chapter 0: Problem 7

(a) write the linear function \(f\) such that it has the indicated function values and (b) sketch the graph of the function. $$ f\left(\frac{1}{2}\right)=-6, f(4)=-3 $$

Short Answer

Expert verified
The equation of the function is \(f(x) = \frac{6}{7}x - \frac{6.42857}\). This would result in a straight line graph with slope \(\frac{6}{7}\) and y-intercept -\frac{6.42857}.

Step by step solution

01

Determine the Slope

The slope of a linear function can be determined using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Given that \(f(\frac{1}{2}) = -6\) and \(f(4) = -3\), this gives two points on the line: (\(\frac{1}{2}, -6) and (4, -3). Using these, the slope is calculated as: \(m = \frac{-3 - (-6)}{4 - \frac{1}{2}} = \frac{3}{\frac{7}{2}} = \frac{6}{7}\).
02

Solve for the Y-intercept

The y-intercept (b) of a linear function can be found using the formula \(b = y - mx\), where m is the slope and the pair (x, y) is any point on the line. Using the point (\(\frac{1}{2}, -6) and the slope \(\frac{6}{7}\), one can calculate the y-intercept as: \(b = -6 - \frac{6}{7}*\frac{1}{2} = -\frac{42}{7} - \frac{3}{7} = -\frac{45}{7} = -\frac{6.42857}\).
03

Write the Equation of the Line

The equation of a line can be written in the form \(y = mx + b\), where m is the slope and b is the y-intercept. Using the slope of \(\frac{6}{7}\) and y-intercept of -\frac{6.42857}, the equation of the function f(x) can be written as \(f(x) = \frac{6}{7}x - \frac{6.42857}\).
04

Sketch the Graph

For sketching the graph, mark the y-intercept on the y-axis at -\frac{6.42857} and the rise-over-run using the slope of \(\frac{6}{7}\). First get up 6 units (as positive 6 is the numerator) and then move to the right by 7 units (as 7 is the denominator of the slope) to mark another point on the graph. Then use a ruler to draw a straight line passing through the marked points.

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Key Concepts

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

Slope Calculation
To find the slope of a linear function, we use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula is all about determining how steep the line is. In simpler terms, the slope tells you how much the function value (or \( y \)) changes for a given change in \( x \).
For example, in our exercise, we have two points: \( \left( \frac{1}{2}, -6 \right) \) and \( (4, -3) \). The slope \( m \) can be calculated by plugging these points into the formula:
  • Subtract the \( y \)-values: \( -3 - (-6) = 3 \)
  • Subtract the \( x \)-values: \( 4 - \frac{1}{2} = \frac{7}{2} \)
So, the slope \( m \) is \( \frac{3}{\frac{7}{2}} = \frac{6}{7} \). A positive slope like \( \frac{6}{7} \) indicates that the function increases as \( x \) increases. This means the line will slant upwards from left to right.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. It鈥檚 denoted as \( b \) in the equation of a line \( y = mx + b \). To find \( b \), you can use the formula \( b = y - mx \), where \( m \) is the slope, and \( (x, y) \) is any known point on the line.
Given one of our points \( \left( \frac{1}{2}, -6 \right) \), we can substitute \( m = \frac{6}{7} \) and solve for \( b \):
  • Start with \( -6 - \frac{6}{7} \times \frac{1}{2} \)
  • Calculate \( b \): \( -\frac{42}{7} - \frac{3}{7} = -\frac{45}{7} \)
  • \( b \) is approximately \( -6.42857 \)
This means the line crosses the y-axis just under -6.5. The y-intercept helps in understanding where your function starts on the y-axis.
Graphing Linear Equations
Graphing linear equations is like connecting dots on a plane to visualize a function. Start by plotting the y-intercept on the y-axis. For our function, this is \( b = -6.42857 \), which is slightly below -6.5 on the y-axis.
Next, use the slope \( \frac{6}{7} \) to determine another point. Move up (rise) by 6 units from the y-intercept, and then to the right (run) by 7 units. Mark this second point.
  • Begin by plotting the point at \( (0, -6.42857) \)
  • From this point, move up 6, right 7 to find another point
  • Draw a line through these points with a ruler for precision
Make sure your graph is neat to accurately represent the equation. This visual helps to quickly see how the function behaves.
Writing Linear Equations
Writing linear equations in the form \( y = mx + b \) is putting together the slope and y-intercept to describe the function completely. Once you have these two values, assembling the equation is straightforward. For our exercise, we found:
  • The slope \( m = \frac{6}{7} \)
  • The y-intercept \( b = -6.42857 \)
So, the equation of the line can be written as: \( y = \frac{6}{7}x - 6.42857 \).
This equation tells us that for any value of \( x \), we can find \( y \) by multiplying \( x \) by \( \frac{6}{7} \) and then subtracting 6.42857. Practicing writing these equations helps in understanding how changes in the slope or y-intercept affect the graph.

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Most popular questions from this chapter

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{2}, \quad x \geq 0 $$

Transportation For groups of 80 or more people, a charter bus company determines the rate per person according to the formula Rate \(=8-0.05(n-80), \quad n \geq 80\) where the rate is given in dollars and \(n\) is the number of people. (a) Write the revenue \(R\) for the bus company as a function of \(n\). (b) Use the function in part (a) to complete the table. What can you conclude? \begin{tabular}{|l|l|l|l|l|l|l|l|} \hline\(n\) & 90 & 100 & 110 & 120 & 130 & 140 & 150 \\ \hline\(R(n)\) & & & & & & & \\ \hline \end{tabular}

In Exercises 33-38, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$ f(x)=10 $$

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