Applied Algebra: Modeling and functions

15.

1. Use your calculator to graph the equations \(y=3x+8\) and \(y=3.1x+6\) together in the standard window. Do you think the lines are parallel?
2. Find the slope of each line in part (a). Are the lines parallel?
3. Find the \(y\)-value for each equation when \(x=20\text{.}\) What do your answers tell you about the two lines?

16.

1. Use your calculator to graph the equation \(y=0.001x+4\) in the standard window. Do you think the line is horizontal?
2. Find the slope and the \(x\)-intercept of the line in part (a). Is the line horizontal?

3. Graph the equation in part (a) in the window

   \(\text{Xmin}=-5000\text{; Xmax}=5000\)

   \(\text{Ymin}=-10\text{; Ymax}=10 \)

   Find the coordinates of two convenient points on the line, and compute its slope using the slope formula.

17.

The slopes of several lines are given below. Which of the lines are parallel to the graph of \(y=0.75x+2\text{,}\) and which are perpendicular to it?

1. \(\displaystyle m=\dfrac{3}{4}\)
2. \(\displaystyle m=\dfrac{8}{6}\)
3. \(\displaystyle m=\dfrac{-20}{15}\)
4. \(\displaystyle m=\dfrac{-39}{52}\)
5. \(\displaystyle m=\dfrac{4}{3}\)
6. \(\displaystyle m=\dfrac{-16}{12}\)
7. \(\displaystyle m=\dfrac{36}{48}\)
8. \(\displaystyle m=\dfrac{9}{12}\)

18.

The slopes of several lines are given below. Which of the lines are parallel to the graph of \(y=2.5x-3\text{,}\) and which are perpendicular to it?

1. \(\displaystyle m=\dfrac{2}{5}\)
2. \(\displaystyle m=\dfrac{25}{10}\)
3. \(\displaystyle m=\dfrac{-8}{20}\)
4. \(\displaystyle m=\dfrac{-45}{18}\)
5. \(\displaystyle m=\dfrac{40}{16}\)
6. \(\displaystyle m=25\)
7. \(\displaystyle m=\dfrac{-1}{25}\)
8. \(\displaystyle m=\dfrac{-5}{10}\)

19.

In each part, determine whether the two lines are parallel, perpendicular, or neither.

1. \(\displaystyle y=\dfrac{3}{5}x-7,~~3x-5y=2\)
2. \(\displaystyle y=4x+3,~~y=\dfrac{1}{4}x-3\)
3. \(\displaystyle 6x+2y=1,~~x=1-3y\)
4. \(\displaystyle 2y=5,~~5y=-2\)

20.

In each part, determine whether the two lines are parallel, perpendicular, or neither.

1. \(\displaystyle 2x-7y=14,~~7x-2y=14\)
2. \(\displaystyle x+y=6,~~x-y=6\)
3. \(\displaystyle x=-3,~~3y=5\)
4. \(\displaystyle \dfrac{1}{4}x-\dfrac{3}{4}y=\dfrac{2}{3},~~\dfrac{1}{6}x+\dfrac{1}{2}y=\dfrac{1}{3}\)

21.

1. Sketch the triangle with vertices \(A(2,5),~B(5,2),\) and \(C(10,7)\text{.}\)
2. Show that the triangle is a right triangle. (Hint: What should be true about the slopes of the two sides that form the right angle?)

22.

1. Sketch the triangle with vertices \(P(-1,3),~Q(-3,8),\) and \(R(4,5)\text{.}\)
2. Show that the triangle is a right triangle. (See the hint for Problem 21.)

23.

1. Sketch the quadrilateral with vertices \(P(2,4),~Q(3,8),~R(5,1),\) and \(S(4,-3)\text{.}\)
2. Show that the quadrilateral is a parallelogram. (Hint: What should be true about the slopes of the opposite sides of the parallelogram?)

24.

1. Sketch the quadrilateral with vertices \(A(-5,4),~B(7,-11),~C(12,25),\) and \(D(0,40)\text{.}\)
2. Show that the quadrilateral is a parallelogram. (See the hint for Problem 23.)

25.

Show that the line passing through the points \(A(0,-3)\) and \(B(3,\dfrac{1}{2})\) also passes through the point \(C(-6,-10)\text{.}\)

26.

Do the points \(P(-5,-3\dfrac{1}{2}),~Q(4,-2)\) and \(R(9\dfrac{1}{2},-1)\) lie on the same line? Why or why not?

31.

Two of the vertices of rectangle \(ABCD\) are \(A(-5,2)\) and \(B(-2,4)\text{.}\)

1. Find an equation for the line that includes side \(\overline{AB}\text{.}\)
2. Find an equation for the line that includes side \(\overline{BC}\) .

32.

Two of the vertices of rectangle \(PQRS\) are \(P(-2,-6)\) and \(Q(4,-4)\text{.}\)

1. Find an equation for the line that includes side \(\overline{PQ}\text{.}\)
2. Find an equation for the line that includes side \(\overline{QR}\text{.}\)

37.

In this exercise we will show that parallel lines have the same slope. In the figure below, \(l_1\) and \(l_2\) are two parallel lines that are neither horizontal nor vertical. Their \(y\)-intercepts are \(A\) and \(B\text{.}\) The segments \(\overline{AC}\) and \(\overline{CD}\) are constructed parallel to the \(x\)-and \(y\)-axes, respectively.

Explain why each of the following statements is true.

1. Angle \(ACD\) equals angle \(CAB\text{.}\)
2. Angle \(DAC\) equals angle \(ACB\text{.}\)
3. Triangle \(ACD\) is similar to triangle \(CAB\text{.}\)
4. \(\displaystyle m_1=\dfrac{CD}{AC};~~m_2=\dfrac{AB}{AC}\)
5. \(\displaystyle m_1=m_2\)

38.

In this exercise we will show that if two lines with slopes \(m_1\) and \(m_2\) (where neither line is vertical) are perpendicular, then \(m_2\) is the negative reciprocal of \(m_1\text{.}\) In the figure below, lines \(l_1\) and \(l_2\) are perpendicular. Their \(y\)-intercepts are \(B\) and \(C\text{.}\) The segment \(\overline{AP}\) is constructed through the point of intersection of \(l_1\) and \(l_2\) parallel to the \(x\)-axis.

Explain why each of the following statements is true.

1. Angles \(ABC\) and \(ACB\) are complementary.
2. Angles \(ABC\) and \(BAP\) are complementary.
3. Angle \(BAP\) equals angle \(ACB\text{.}\)
4. Angles \(CAP\) and \(ACB\) are complementary.
5. Angle \(CAP\) equals angle \(ABC\text{.}\)
6. Triangle \(ABP\) is similar to triangle \(CAP\text{.}\)
7. \(\displaystyle m_1=\dfrac{BP}{AP};~~m_2=-\dfrac{CP}{AP}\)
8. \(\displaystyle m_2=\dfrac{-1}{m_1}\)