Chapter 7

\(y \geq-2 x-1\)

\(x=-y^{2}+5\)

\(2 x-y \geq 0\)

Verify that the points \((-3,1),(5,7)\), and \((8,3)\) are vertices of a right triangle. [Hint: If \(a^{2}+b^{2}=c^{2}\), then it is a right triangle with the right angle opposite side \(c .]\)

\(-x+4 y-4 \leq 0\)

Verify that the points \((0,3),(2,-3)\), and \((-4,-5)\) are vertices of an isosceles triangle.

Verify that the points \((7,12)\) and \((11,18)\) divide the line segment joining \((3,6)\) and \((15,24)\) into three segments of equal length.

Verify that \((3,1)\) is the midpoint of the line segment joining \((-2,6)\) and \((8,-4)\).

\(x \leq 3\)

\(x \leq-1\) and \(y<1\)

\(x^{2}+y^{2}-4 x-12=0\)

Graph \(|x|<2\). [Hint: Remember that \(|x|<2\) is equivalent to \(-2

\(\operatorname{Graph}|y|>1\).

\(x\) intercept of \(-3\) and slope of \(-\frac{5}{8}\)

Find \(x\) if the line through \((-2,4)\) and \((x, 6)\) has a slope of \(\frac{2}{9}\).

Graph \(|x+y|<1\).

Find \(y\) if the line through \((1, y)\) and \((4,2)\) has a slope of \(\frac{5}{3}\).

Contains the point \((2,-4)\) and is parallel to the \(y\) axis

Find \(x\) if the line through \((x, 4)\) and \((2,-5)\) has a slope of \(-\frac{9}{4}\)

Contains the point \((-3,-7)\) and is parallel to the \(x\) axis

Find \(y\) if the line through \((5,2)\) and \((-3, y)\) has a slope of \(-\frac{7}{8}\)

Contains the point \((5,6)\) and is perpendicular to the \(y\) axis

Use the DRAW feature of your graphing calculator to draw each of the following. (a) A line segment between \((-2,-4)\) and \((-2,5)\) (b) A line segment between \((2,2)\) and \((5,2)\) (c) A line segment between \((2,3)\) and \((5,7)\) (d) A triangle with vertices at \((1,-2),(3,4)\), and \((-3,6)\)

-3 y=-x+3 $$

Contains the point \((-4,7)\) and is perpendicular to the \(x\) axis

Suppose that the daily profit from an ice cream stand is given by the equation \(p=2 n-4\), where \(n\) represents the number of gallons of ice cream mix used in a day, and \(p\) represents the number of dollars of profit. Label the horizontal axis \(n\) and the vertical axis \(p\), and graph the equation \(p=2 n-4\) for nonnegative values of \(n\).

Contains the point \((1,3)\) and is parallel to the line \(x+\) \(5 y=9\)

The cost (c) of playing an online computer game for a time \((t)\) in hours is given by the equation \(c=3 t+5\). Label the horizontal axis \(t\) and the vertical axis \(c\), and graph the equation for nonnegative values of \(t\).

An online grocery store charges for delivery based on the equation \(C=0.30 p\), where \(C\) represents the cost in dollars, and \(p\) represents the weight of the groceries in pounds. Label the horizontal axis \(p\) and the vertical axis \(C\), and graph the equation \(C=0.30 p\) for nonnegative values of \(p\).

(a) The equation \(\mathrm{F}=\frac{9}{5} \mathrm{C}+32\) can be used to convert from degrees Celsius to degrees Fahrenheit. Complete the following table. \begin{tabular}{l|llllllllll} \(\mathrm{C}\) & 0 & 5 & 10 & 15 & 20 & \(-5\) & \(-10\) & \(-15\) & \(-20\) & \(-25\) \\ \hline \(\mathrm{F}\) & & & & & & & & & & \end{tabular} (b) Graph the equation \(\mathrm{F}=\frac{9}{5} \mathrm{C}+32\). (c) Use your graph from part (b) to approximate values for \(\mathrm{F}\) when \(\mathrm{C}=25^{\circ}, 30^{\circ},-30^{\circ}\), and \(-40^{\circ}\). (d) Check the accuracy of your readings from the graph in part (c) by using the equation \(\mathrm{F}=\frac{9}{5} \mathrm{C}+32\).

Contains the point \((-1,3)\) and is perpendicular to the line \(2 x-y=4\)

(a) Digital Solutions charges for help-desk services according to the equation \(c=0.25 m+10\), where \(c\) represents the cost in dollars, and \(m\) represents the minutes of service. Complete the following table. \begin{tabular}{|l|l|l|l|l|l|l|} \hline\(m\) & 5 & 10 & 15 & 20 & 30 & 60 \\ \hline\(c\) & & & & & & \\ \hline \end{tabular} (b) Label the horizontal axis \(m\) and the vertical axis \(c\), and graph the equation \(c=0.25 m+10\) for nonnegative values of \(m\). (c) Use the graph from part (b) to approximate values for \(c\) when \(m=25,40\), and 45 . (d) Check the accuracy of your readings from the graph in part (c) by using the equation \(c=0.25 m+10\).

Contains the point \((-2,-3)\) and is perpendicular to the line \(x+4 y=6\)

How do we know that the graph of \(y=-3 x\) is a straight line that contains the origin?

How do we know that the graphs of \(2 x-3 y=6\) and \(-2 x+3 y=-6\) are the same line?

What is the graph of the conjunction \(x=2\) and \(y=4\) ? What is the graph of the disjunction \(x=2\) or \(y=4\) ? Explain your answers.

Your friend claims that the graph of the equation \(x=2\) is the point \((2,0)\). How do you react to this claim?

\(|x+y|=1\)

\(y=\frac{-1}{x^{2}}\)

\(|x-y|=4\)

\(|2 x-y|=4\)

(a) Graph \(y=3 x+4, y=2 x+4, y=-4 x+4\), and \(y=-2 x+4\) on the same set of axes. (b) Graph \(y=\frac{1}{2} x-3, y=5 x-3, y=0.1 x-3\), and \(y=-7 x-3\) on the same set of axes. (c) What characteristic do all lines of the form \(y=\) \(a x+2\) (where \(a\) is any real number) share?

(a) Graph \(y=2 x-3, y=2 x+3, y=2 x-6\), and \(y=\) \(2 x+5\) on the same set of axes. (b) Graph \(y=-3 x+1, y=-3 x+4, y=-3 x-2\), and \(y=-3 x-5\) on the same set of axes. (c) Graph \(y=\frac{1}{2} x+3, y=\frac{1}{2} x-4, y=\frac{1}{2} x+5\), and \(y=\frac{1}{2} x-2\) on the same set of axes. (d) What relationship exists among all lines of the form \(y=3 x+b\), where \(b\) is any real number?

(a) Graph \(2 x+3 y=4,2 x+3 y=-6,4 x-6 y=7\), and \(8 x+12 y=-1\) on the same set of axes. (b) Graph \(5 x-2 y=4,5 x-2 y=-3,10 x-4 y=3\), and \(15 x-6 y=30\) on the same set of axes. (c) Graph \(x+4 y=8,2 x+8 y=3, x-4 y=6\), and \(3 x+12 y=10\) on the same set of axes. (d) Graph \(3 x-4 y=6,3 x+4 y=10,6 x-8 y=20\), and \(6 x-8 y=24\) on the same set of axes. (e) For each of the following pairs of lines, (a) predict whether they are parallel lines, and (b) graph each pair of lines to check your prediction. (1) \(5 x-2 y=10\) and \(5 x-2 y=-4\) (2) \(x+y=6\) and \(x-y=4\) (3) \(2 x+y=8\) and \(4 x+2 y=2\) (4) \(y=0.2 x+1\) and \(y=0.2 x-4\) (5) \(3 x-2 y=4\) and \(3 x+2 y=4\) (6) \(4 x-3 y=8\) and \(8 x-6 y=3\) (7) \(2 x-y=10\) and \(6 x-3 y=6\) (8) \(x+2 y=6\) and \(3 x-6 y=6\)

Now let's use a graphing calculator to get a graph of \(\mathrm{C}=\frac{5}{9}(\mathrm{~F}-32)\). By letting \(\mathrm{F}=x\) and \(\mathrm{C}=y\), we obtain Figure 7.15. Pay special attention to the boundaries on \(x\). These values were chosen so that the fraction \(\frac{\text { (Maximum value of } x \text { ) minus (Minimum value of } x \text { ) }}{95}\) would be equal to 1 . The viewing window of the graphing calculator used to produce Figure \(7.15\) is 95 pixels (dots) wide. Therefore, we use 95 as the denominator of the fraction. We chose the boundaries for \(y\) to make sure that the cursor would be visible on the screen when we looked for certain values. Now let's use the TRACE feature of the graphing calculator to complete the following table. Note that the cursor moves in increments of 1 as we trace along the graph. \begin{tabular}{l|lllllllll} F & \(-5\) & 5 & 9 & 11 & 12 & 20 & 30 & 45 & 60 \\ \hline C & & & & & & & & & \end{tabular} (This was accomplished by setting the aforementioned fraction equal to 1 .) By moving the cursor to each of the F values, we can complete the table as follows. \begin{tabular}{r|rrrrrrrrr} F & \(-5\) & 5 & 9 & 11 & 12 & 20 & 30 & 45 & 60 \\ \hline C & \(-21\) & \(-15\) & \(-13\) & \(-12\) & \(-11\) & \(-7\) & \(-1\) & 7 & 16 \end{tabular} The \(\mathrm{C}\) values are expressed to the nearest degree. Use your calculator and check the values in the table by using the equation \(\mathrm{C}=\frac{5}{9}(\mathrm{~F}-32)\).

\(y=\frac{-2}{x^{2}+1}\)

A certain highway has a \(2 \%\) grade. How many feet does it rise in a horizontal distance of 1 mile? (1 mile \(=\) 5280 feet)

The grade of a highway up a hill is \(30 \%\). How much change in horizontal distance is there if the vertical height of the hill is 75 feet?

How would you convince someone that there are infinitely many ordered pairs of real numbers that satisfy \(x+y=7 ?\)

Suppose that a highway rises a distance of 215 feet in a horizontal distance of 2640 feet. Express the grade of the highway to the nearest tenth of a percent.