Stewart Precalc 6e Section 1.10: Lines

1 Concept - Slope Formula · Level 1

We find the steepness, or slope, of a line passing through two points by dividing the difference in the ___-coordinates of these points by the difference in the ___-coordinates. So the line passing through the points \((0, 1)\) and \((2, 5)\) has slope ___.

2 Concept - Slope, Parallel, and Perpendicular · Level 1

A line has the equation \(y = 3 x - 2\).

(a) This line has slope ___.

Answer for 2(a)

(b) Any line parallel to this line has slope ___.

Answer for 2(b)

(c) Any line perpendicular to this line has slope ___.

Answer for 2(c)

Enter your answer directly below each part above.

3 Concept - Point-Slope Form · Level 1

The point-slope form of the equation of the line with slope \(3\) passing through the point \((1, 2)\) is ___.

4 Concept - Horizontal and Vertical Lines · Level 1

(a) The slope of a horizontal line is ___. The equation of the horizontal line passing through \((2, 3)\) is ___.

Answer for 4(a)

(b) The slope of a vertical line is ___. The equation of the vertical line passing through \((2, 3)\) is ___.

Answer for 4(b)

Enter your answer directly below each part above.

5 Skills - Slope Through Two Points · Level 1

Find the slope of the line through \(P\) and \(Q\), where \(P(0, 0)\) and \(Q(4, 2)\).

6 Skills - Slope Through Two Points · Level 1

Find the slope of the line through \(P\) and \(Q\), where \(P(0, 0)\) and \(Q(2, -6)\).

7 Skills - Slope Through Two Points · Level 1

Find the slope of the line through \(P\) and \(Q\), where \(P(2, 2)\) and \(Q(-10, 0)\).

8 Skills - Slope Through Two Points · Level 1

Find the slope of the line through \(P\) and \(Q\), where \(P(1, 2)\) and \(Q(3, 3)\).

9 Skills - Slope Through Two Points · Level 1

Find the slope of the line through \(P\) and \(Q\), where \(P(2, 4)\) and \(Q(4, 3)\).

10 Skills - Slope Through Two Points · Level 1

Find the slope of the line through \(P\) and \(Q\), where \(P(2, -5)\) and \(Q(-4, 3)\).

11 Skills - Slope Through Two Points · Level 1

Find the slope of the line through \(P\) and \(Q\), where \(P(1, -3)\) and \(Q(-1, 6)\).

12 Skills - Slope Through Two Points · Level 1

Find the slope of the line through \(P\) and \(Q\), where \(P(-1, -4)\) and \(Q(6, 0)\).

13 Exercise - Discussion: collinearity methods · Level 2

Collinear Points. Suppose that you are given the coordinates of three points in the plane and you want to see whether they lie on the same line. How can you do this using slopes? Using the Distance Formula? Can you think of another method?

14 Example - Finding the Slope of a Line Through Two Points · Level 1

Find the slope of the line that passes through the points \(P(2, 1)\) and \(Q(8, 5)\).

15 Example - Finding the Equation of a Line with Given Point and Slope · Level 2

(a) Find an equation of the line through \((1, -3)\) with slope \(-\dfrac{1}{2}\). (b) Sketch the line.

16 Example - Finding the Equation of a Line Through Two Given Points · Level 2

Find an equation of the line through the points \((-1, 2)\) and \((3, -4)\).

17 Example - Lines in Slope-Intercept Form · Level 2

(a) Find the equation of the line with slope 3 and \(y\)-intercept \(-2\). (b) Find the slope and \(y\)-intercept of the line \(3y - 2x = 1\).

18 Example - Vertical and Horizontal Lines · Level 1

(a) Write an equation for the vertical line through \((3, 5)\). (b) Describe the graph of the equation \(x = 3\). (c) Write an equation for the horizontal line through \((8, -2)\). (d) Describe the graph of the equation \(y = -2\).

19 Example - Graphing a Linear Equation · Level 2

Sketch the graph of the equation \(2 x - 3 y - 12 = 0\).

20 Example - Equation of a Parallel Line · Level 3

Find an equation of the line through the point \((5, 2)\) that is parallel to the line \(4 x + 6 y + 5 = 0\).

21 Example - Perpendicular Lines and Right Triangle · Level 3

Show that the points \(P(3, 3)\), \(Q(8, 17)\), and \(R(11, 5)\) are the vertices of a right triangle.

22 Example - Equation of a Perpendicular Line · Level 3

Find an equation of the line that is perpendicular to the line \(4 x + 6 y + 5 = 0\) and passes through the origin.

23 Example - Graphing a Family of Lines · Level 2

Use a graphing calculator to graph the family of lines \(y = 0.5 x + b\) for \(b = -2, -1, 0, 1, 2\). What property do the lines share?

24 Example - Slope as Rate of Change (Reservoir) · Level 3

A dam is built on a river to create a reservoir. The water level \(w\) in the reservoir is given by the equation \(w = 4.5 t + 28\), where \(t\) is the number of years since the dam was constructed and \(w\) is measured in feet.

(a) Sketch a graph of this equation.

Answer for 24(a)

(b) What do the slope and the \(w\)-intercept of this graph represent?

Answer for 24(b)

Enter your answer directly below each part above.

25 Example - Linear Model for Temperature and Elevation · Level 3

As dry air moves upward, it expands and cools. The ground temperature is 20°C and the temperature at a height of 1 km is 10°C. Assume the relationship between temperature \(T\) (in °C) and height \(h\) (in km) is linear.

(a) Express \(T\) in terms of \(h\).

Answer for 25(a)

(b) Draw the graph of the linear equation. What does its slope represent?

Answer for 25(b)

(c) What is the temperature at a height of \(2.5\) km?

Answer for 25(c)

Enter your answer directly below each part above.

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