1. 1.4 Lines
Essential Question: How can you use the equations of two non-vertical lines to tell whether the lines are parallel or perpendicular?

2. An arithmetic sequence is nothing more than a linear equation…
{3, 8, 13, 18, … } → un = 3 + (n - 1)(5) → un = 3 + 5n – → un = 5n – 2
If we replace n with x and un with y, we have the linear equation: y = 5x – 2
The sequence above corresponds to the points { (1, 3), (2, 8), (3, 13), (4, 18), … }
1.4 Lines

3. 1.4 Lines Slope Example 1 Change in y ÷ change in x
Δ y ÷ Δx (delta y ÷ delta x)
Example 1
Find the slope of the line that passes through (0,-1) and (4,1)
1.4 Lines

4. 1.4 Lines Finding Slope From a Graph Properties of Slope
Find two points on the coordinate plane, and use the slope formula
Properties of Slope
If m > 0, the line rises from left to right. The larger m is, the more steeply the line rises.
If m = 0, the line is horizontal
If m < 0, the line falls from left to right. The larger m is, the more steeply the line falls.
1.4 Lines

5. 1.4 Lines Slope-Intercept Form
y = mx + b
“m” is the slope
“b” is the y-intercept (the point where the graph crosses the y-axis)
Example 3: Graphs of Arithmetic Sequences
1st three terms of an arithmetic sequence are -2, 3, and 8. Use an explicit function and compare it to slope-intercept form
1.4 Lines

6. 1.4 Lines Example 3 (Continued) Sequence: -2, 3, 8, …
Explicit form: un = u1 + (n-1)(d)
u1 = , d =
What connection(s) do you see between the slope-intercept form and the original sequence? -2 5
un = -2 + (n-1)(5) un = n – 5 un = 5n – 7
1.4 Lines

7. 1.4 Lines Connection between Arithmetic Sequences and Lines
Explicit Form: un = u1 + (n-1)(d)
Slope Intercept Form: y = mx + b
The slope corresponds to the common difference (m = d)
The y-intercept represents the value of u0, or the term before the sequence started, which is u1 less one common difference (u1 – d)
1.4 Lines

8. 1.4 Lines Graphing a Line Graphing calculator
Solve an equation for y (i.e. get y by itself)
Plot the y-intercept
Use the slope (rise over run) to make a 2nd plot
Draw a line which connects the two dots (a line, not a segment)
Graphing calculator
Graph → F1, input in the equation, solved for y
2nd, F5
1.4 Lines

9. 1.4 Lines Assignment Page 40 Problems 3 – 14 (all problems)
Show your work
1.4 Lines

10. 1.4 Lines (Day 2)
Essential Question: How can you use the equations of two non-vertical lines to tell whether the lines are parallel or perpendicular?

11. 1.4 Lines Point-Slope Form
Use point slope form when you’re given a point and a slope (SURPRISE!) or two points to determine an equation (and use those points to determine the slope)
y – y1 = m(x – x1)
Any point given can be used for (x1, y1)
1.4 Lines

12. 1.4 Lines Point-Slope Form (Example 6)
Find the equation of the line that passes through the point (1, -6) with a slope 2.
y – y1 = m(x – x1)
y – (-6) = 2(x – 1)
y + 6 = 2x – 2
– – 6
y = 2x – 8
1.4 Lines

13. 1.4 Lines Vertical and Horizontal Lines Equation of a Horizontal Line
Every point along a horizontal line will have the same y value.
Written as y = b (where b is the y-intercept)
The slope of a horizontal line = 0
Equation of a Vertical Line
Every point along a vertical line will have the same x value.
Written as x = c (where c is some constant)
The slope of a vertical line is undefined
1.4 Lines

14. 1.4 Lines Parallel & Perpendicular Lines
Parallel lines have the same slope
Perpendicular lines have inverse reciprocal slopes
That means the product of the slopes of two perpendicular lines is -1
Take the slope of one line, flip as a fraction and flip sign.
1.4 Lines

15. Parallel & Perpendicular Lines (Example 9)
Given line M whose equation is 3x - 2y + 6=0, find the equation of the parallel and perpendicular lines which go through the point (2, -1)
Parallel Line
Get y by itself to find the slope of line M
The slope of M is 3/2
Use point-slope form
y – y1 = m(x – x1)
3x – 2y + 6 = 0
–3x –6 –3x – 6
y – (-1) = 3/2(x – 2)
-2y = -3x – 6
y + 1 = 3/2x – 3
-2 -2 -2
– – 1
y = 3/2x + 3
y = 3/2x – 4

16. Parallel & Perpendicular Lines (Example 9)
Given line M whose equation is 3x - 2y + 6=0, find the equation of the parallel and perpendicular lines which go through the point (2, -1)
Perpendicular Line
Slope of M = 3/2, so the perpendicular slope is -2/3
Use point-slope form
y – y1 = m(x – x1)
y – (-1) = -2/3(x – 2)
y + 1 = -2/3x + 4/3
– – 1
y = -2/3x + 1/3

17. 1.4 Lines Standard Form of a Line
Written as Ax + By = C, where A, B, and C are integers
We can convert our perpendicular line to standard form by removing any fractions
1.4 Lines

18. 1.4 Lines Forms of Linear Equations Standard Form: Ax + By = C
All integers, x and y terms on same side
Slope-Intercept Form: y = mx + b
Best for quickly identifying slope and graphing
Point-Slope Form: y – y1 = m(x – x1)
Best for writing equations
Horizontal Lines: y = b
Slope = 0
Vertical Lines: x = c
Slope is undefined
1.4 Lines

19. 1.4 Lines Assignment Page 40 – 41 Problems 17 – 49 (odd problems)
Show your work
1.4 Lines