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Chapter 2: Problem 54

If a linear function \(f\) satisfles the given conditions, find \(f(x)\). $$f(-2)=7 \text { and } f(4)=-2$$

Short Answer

Expert verified
The function is \(f(x) = -\frac{3}{2}x + 4\).

Step by step solution

01

Understanding the Problem

We need to find the linear function \(f(x)\) that passes through the points \((-2, 7)\) and \((4, -2)\). Linear functions have the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
02

Finding the Slope

The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Here, substituting \((-2, 7)\) and \((4, -2)\) gives: \[ m = \frac{-2 - 7}{4 - (-2)} = \frac{-9}{6} = -\frac{3}{2} \].
03

Using the Point-Slope Form

With the slope \(m = -\frac{3}{2}\) and a point \((-2, 7)\), we use the point-slope form to find the equation of the line: \[ y - y_1 = m(x - x_1) \]. Plugging in the values gives: \[ y - 7 = -\frac{3}{2}(x + 2) \].
04

Converting to Slope-Intercept Form

Expand and simplify the equation: \( y - 7 = -\frac{3}{2}(x + 2) \). This becomes: \[ y - 7 = -\frac{3}{2}x - 3 \]. Adding 7 to both sides, we get \[ y = -\frac{3}{2}x + 4 \]. So, the function \(f(x)\) is \(f(x) = -\frac{3}{2}x + 4\).

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Key Concepts

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

Slope
In a linear function, the **slope** is a crucial concept. It tells you how steep the line is and in which direction it goes. The slope measures the rate of change of the function as you move along the x-axis. It essentially describes the relationship between \(x\) and \(y\) values.The formula to find the slope \(m\) between two points, \((x_1, y_1)\) and \((x_2, y_2)\), is:
  • \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Here is how it works: subtract the \(y\)-values and divide by the subtraction of \(x\)-values. Remember:
  • A positive slope means the line goes upwards as we move right.
  • A negative slope means the line goes downwards.
  • If the slope is zero, the line is horizontal.
  • An undefined slope occurs for a vertical line where \(\Delta x = 0\).
For the points \((-2,7)\) and \((4,-2)\) from the original exercise, the slope is calculated as \(-\frac{3}{2}\), indicating the line declines from left to right.
Y-Intercept
The **y-intercept** is where a line crosses the y-axis. In other words, it is the value of \(y\) when \(x\) is zero. This point is very useful because it helps establish the equation of the line quickly. In the slope-intercept form of a line, \(y = mx + b\), \(b\) represents the y-intercept.When you're given a slope and one point, you can easily determine the full line equation by knowing the y-intercept.For the function \(f(x) = -\frac{3}{2}x + 4\), the y-intercept is
  • \(4\)
This means that regardless of x's value, the line will cross the y-axis at y\(=4\). This is key to quickly sketching or understanding the linear function's graph.
Point-Slope Form
The **point-slope form** is one of the most useful forms of a linear equation, especially when you have a point and a slope. It looks like this:
  • \(y - y_1 = m(x - x_1)\)
This formula is very handy when you're given a point \((x_1, y_1)\) along with the slope \(m\).Using the point-slope form can simplify the process of finding the linear equation's formula. From our problem, we used the point-slope form starting with the slope \(-\frac{3}{2}\) and the point \((-2, 7)\):Replacing into the form gives:
  • \(y - 7 = -\frac{3}{2}(x + 2)\)
This equation reflects the line that passes through the point \((-2,7)\) with the given slope. From here, you can convert to slope-intercept form to find the y = form, making it even easier to graph or analyze.

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Most popular questions from this chapter

Cars are crossing a bridge that is 1 mile long. Each car is 12 feet long and is required to stay a distance of at least \(d\) feet from the car in front of it (see figure). (a) Show that the largest number of cars that can be on the bridge at one time is \(15280 /(12+d)]\), where I I denotes the greatest integer function. (b) If the velocity of each car is \(v\) mi/hr, show that the maximum traffic flow rate \(F\) (in cars/hr) is given by \(F=[5280 v /(12+d)]\)

Graph \(f\) in the viewing rectangle \([-12,12]\) by \([-8,8] .\) Use the graph of \(f\) to predict the graph of \(g .\) Verify your prediction by graphing \(g\) in the same viewing rectangle. $$f(x)=|x+2| ; \quad g(x)=|x-3|-3$$

Ozone occurs at all levels of the earth's atmosphere. The density of ozone varies both seasonally and latitudinally. At Edmonton, Canada, the density \(D(h)\) of ozone \(\left(\text { in } 10^{-3} \mathrm{cm} / \mathrm{km}\right)\) for altitudes \(h\) between \(20 \mathrm{kllo}\) meters and 35 kilometers was determined experimentally. For each \(D(h)\) and season, approximate the altitude at which the density of ozone is greatest. $$D(h)=-0.058 h^{2}+2.867 h-24.239 \text { (autumn) }$$

Several values of two functions \(T\) and \(S\) are listed in the following tables: $$\begin{array}{|c|ccccc|} \hline t & 0 & 1 & 2 & 3 & 4 \\ \hline T(t) & 2 & 3 & 1 & 0 & 5 \\ \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline S(x) & 1 & 0 & 3 & 2 & 5 \\ \hline \end{array}$$ If possible, find (a) \((T \circ S)(1)\) \((s \circ T)(1)\) (c) \((T \circ T)(1)\) (d) \((S \circ S)(1)\) (e) \((T \circ S)(4)\)

If \(f(x)=\sqrt{x}-1\) and \(g(x)=x^{3}+1,\) approximate \(\left(f^{\circ} g\right)(0.0001) .\) In order to avoid calculating a zero value for \((f \circ g)(0.0001),\) rewrite the formula for \(f^{\circ}\) g as $$\frac{x^{3}}{\sqrt{x^{3}+1}+1}$$
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