5.1 writing linear equations in slope intercept form
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You can use this presentation to introduce students in how to write linear equations given the slope and the y-intercept. This is the first case in writing linear equations.
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This document discusses linear functions and slope-intercept form. It defines slope and describes how to calculate the slope between two points. It explains that a linear function is one whose graph is a straight line and can be represented by a linear equation. It defines y-intercepts and x-intercepts as the points where a line crosses the y-axis and x-axis. It shows how to write a linear equation in slope-intercept form using the slope and y-intercept. Examples are given for finding the slope and y-intercept of a line and writing the line's equation in slope-intercept form. It also demonstrates how to graph a line by using its equation in slope-intercept form.
2.5 Equations of Lines
This document discusses equations of lines, including linear functions and their standard, point-slope, and slope-intercept forms. It defines parallel and perpendicular lines based on their slopes. It provides examples of writing the equation of a line from its graph or from a point and slope. Key steps include calculating the slope, selecting a point, and writing the equation in the appropriate form.
4.6 quick graphs using slope intercept form
The following presentation helps teachers and students to work with the slope and y-intercept as tools for graphing. It also shows the relationship between parallel lines and the slopes of the lines.
5 8 Parallel Perpendicular Lines
This document provides background information and examples on parallel and perpendicular lines. It discusses how to identify the slope of a line from its equation in slope-intercept or standard form. It notes that parallel lines have the same slope, while perpendicular lines have slopes that are opposite reciprocals whose product is -1. Examples are provided to find the slopes of parallel and perpendicular lines and to write equations of lines given a point and slope.
2.4
This document discusses linear equations and their various forms. It covers finding the equation of a line given a point and slope using point-slope form, or given two points using the formula for slope and point-slope form. It also discusses writing equations in standard and slope-intercept form, finding intercepts to graph lines, and writing equations of lines parallel or perpendicular to given lines through a specified point. Examples are provided for each concept.
2.4
This document discusses linear equations and their various forms. It covers finding the equation of a line given a point and slope using point-slope form, or given two points using the formula for slope and point-slope form. It also discusses writing equations in standard and slope-intercept form, finding intercepts to graph lines, and writing equations of lines parallel or perpendicular to given lines through a specified point. Examples are provided for each concept.
Calculus
This lecture discusses inequalities, absolute values, and graphs of solutions to inequalities on a coordinate plane. Key points covered include:
- Inequalities relate values that are different using symbols like < and >. Operations on inequalities follow rules like adding/subtracting a positive number or multiplying/dividing by a negative number requires changing the inequality symbol.
- Absolute value expressions use | | to represent the distance from zero. Operations inside | | follow different rules than normal order of operations.
- Graphing solutions to inequalities involves open/closed circles and drawing lines left/right to represent <, >, ≤, or ≥. Examples are worked through to demonstrate graphing linear and quadratic inequalities.
1st trimester exam coverage
This document provides a summary of topics covered in the 1st trimester of Grade 11 Mathematics, including:
1) Classifying polygons using properties like side lengths, slopes, and angles
2) Distinguishing between functions and relations based on their domains and ranges
3) Writing equations to represent loci like lines, circles, and parabolas
4) Calculating directed distance to determine if a point lies on, above, or below a line
5) Finding the intersection of lines algebraically or by checking if a point satisfies both line equations
6) Recognizing circle equations in standard and general form and writing circle equations from graphs
7) Describing parabolas as loci of points equid
Lesson 3-7 Equations of Lines in the Coordinate Plane 189.docx
Lesson 3-7 Equations of Lines in the Coordinate Plane 189
3-7
Objective To graph and write linear equations
Ski resorts often use steepness to rate the difficulty
of their hills. The steeper the hill, the higher the
difficulty rating. Below are sketches of three new hills
at a particular resort. Use each rating level only once.
Which hill gets which rating? Explain.
Difficulty Ratings
Easiest
Intermediate
Difficult
3300 ft 3000 ft 3500 ft
1190 ft 1180 ft 1150 ft
A B C
Th e Solve It involves using vertical and horizontal distances to determine steepness.
Th e steepest hill has the greatest slope. In this lesson you will explore the concept of
slope and how it relates to both the graph and the equation of a line.
Essential Understanding You can graph a line and write its equation when you
know certain facts about the line, such as its slope and a point on the line.
Equations of Lines in the
Coordinate Plane
Think back!
What did you
learn in algebra
that relates to
steepness?
Key Concept Slope
Defi nition
Th e slope m of a line is the
ratio of the vertical change
(rise) to the horizontal change
(run) between any two points.
Symbols
A line contains the
points (x1, y1) and
(x2, y2) .
m 5
rise
run 5
y2 2 y1
x2 2 x1
Diagram
(x2, y2)
(x1, y1)
O
x
y run
rise
Lesson
Vocabulary
• slope
• slope-intercept
form
• point-slope form
L
V
L
V
• s
LL
VVV
• s
hsm11gmse_NA_0307.indd 189 2/24/09 6:49:09 AM
http://media.pearsoncmg.com/aw/aw_mml_shared_1/copyright.html
Problem 1
Got It?
190 Chapter 3 Parallel and Perpendicular Lines
Finding Slopes of Lines
A What is the slope of line b?
m 5
2 2 (22)
21 2 4
5 4
25
5 245
B What is the slope of line d?
m 5
0 2 (22)
4 2 4
5 20 Undefi ned
1. Use the graph in Problem 1.
a. What is the slope of line a?
b. What is the slope of line c?
As you saw in Problem 1 and Got It 1 the slope of a line can be positive, negative,
zero, or undefi ned. Th e sign of the slope tells you whether the line rises or falls
to the right. A slope of zero tells you that the line is horizontal. An undefi ned slope
tells you that the line is vertical.
You can graph a line when you know its equation. Th e equation of a line has diff erent
forms. Two forms are shown below. Recall that the y-intercept of a line is the
y-coordinate of the point where the line crosses the y-axis.
O
y
c
b
d a
x
2
2 86
4
6
( 1, 2)
(4, 2)
(4, 0)
(5, 7)(1, 7)
(2, 3)
Positive slope
O
x
y
Negative slope
O
x
y
Zero slope
O
x
y
Undefined slope
O
x
y
Key Concept Forms of Linear Equations
Defi nition
Th e slope-intercept form of an equation of
a nonvertical line is y 5 mx 1 b, where m
is the slope and b is the y-intercept.
Symbols
Th e point-slope form of an equation of a
nonvertical line is y 2 y1 5 m(x 2 x1),
where m is the slope and (x1, y1) is a point
on the line.
y mx b
slope y-intercept
y y1 m(x x1)
slope x-coordinatey-coordinate
A
...
Slope and Equations of Lines
This document discusses slopes and equations of lines. It defines slope as the rate of change of a straight line and provides the slope formula. Positive slopes mean the graph lines go up from left to right, while negative slopes mean lines go down. There are three common forms for writing linear equations: slope-intercept form where the slope is the coefficient of x; standard form where A, B, and C don't represent slope or y-intercept; and point-slope form used when a point and slope on the line are known. Special cases for horizontal and vertical lines are also covered. Videos demonstrate calculating slope and writing equations in slope-intercept and point-slope form.
4.1 quadratic functions and transformations
This document discusses quadratic functions and transformations. It defines key terms like parabola, vertex, and axis of symmetry. It explains how the a value in the vertex form y=a(x-h)^2+k determines if a parabola is vertically stretched or compressed. It also states that if a is negative, the graph is reflected over the x-axis. The minimum or maximum value of a quadratic is always the y-coordinate of the vertex. The document provides examples of graphing and writing quadratic functions using vertex form.
Chapter 5 Slopes of Parallel and Perpendicular Lines
This document provides a lesson on identifying and writing equations for parallel and perpendicular lines. It includes examples of identifying parallel lines based on their having the same slope. Perpendicular lines are defined as having slopes whose product is -1. The document also demonstrates how to write equations for lines parallel or perpendicular to a given line based on their point and slope. Examples are provided to illustrate finding slopes from graphs and using them to determine if lines are parallel or perpendicular.
Geometry unit 3.8
This document provides examples and explanations about parallel and perpendicular lines. It defines parallel lines as lines in the same plane that do not intersect. Perpendicular lines intersect to form a 90 degree angle. Several examples show how to identify parallel and perpendicular lines based on their slopes. The document also demonstrates applications to geometry, such as showing when a figure like a parallelogram or right triangle is formed based on the relationships between line slopes.
Curves
This document discusses different ways to mathematically represent curves, including polynomial representations and parametric forms. It focuses on cubic polynomials and parametric representations, explaining that parametric form solves problems with explicit and implicit forms by allowing representation of curves with infinite slopes or multiple y-values for a given x-value. Parametric form also makes it easier to combine curve segments continuously. The document then discusses spline curves, which use piecewise cubic polynomial functions to fit smooth curves through points, and cubic splines specifically, providing the equations used to define cubic splines.
5.5 parallel and perpendicular lines (equations) day 1
The document discusses writing equations of parallel and perpendicular lines, including identifying whether lines are parallel or perpendicular based on their slopes. It provides examples of writing equations of parallel and perpendicular lines given a point and the slope or line of another line. Guidance problems are also included to practice these skills.
Geometry unit 3.8
The document discusses parallel and perpendicular lines. It provides examples of identifying parallel and perpendicular lines based on their slopes. It also discusses writing equations of lines parallel and perpendicular to given lines, and applications involving parallel and perpendicular lines in geometry problems involving shapes like parallelograms, rectangles, and right triangles.
Magtibay buk bind#2
The document discusses Hamiltonian graphs, which are graphs that contain a Hamiltonian circuit or path. A Hamiltonian circuit visits each vertex in the graph exactly once and forms a cycle. The document provides an example of a graph that is Hamiltonian but not Eulerian, and vice versa. It notes that while it is clear only connected graphs can be Hamiltonian, there is no simple criterion to determine if a graph is Hamiltonian like there is for Eulerian graphs. It presents Dirac's sufficient condition for a graph to be Hamiltonian.
Intermediate Algebra Question #1 30 pointsGiven the points .docx
Intermediate Algebra
Question #1: 30 points
Given the points A(-5,2), B(3,-4), and C(5,3).
1. Plot the 3 points on a piece of graph paper. Do this by hand. If you use a graphing calculator, copy the graph by hand onto a sheet of graph paper. Electronic plots are not accepted. Remember, you need to know what you are doing so that if you punch in a wrong number, you will know that something is wrong!
2. Draw the line AB on your graph paper.
3. What is the slope of the line AB?
4. What is the y-intercept of the line AB?
5. What is the equation of the line AB in slope-intercept form?
6. What is the slope of the line which passes through point C and is parallel to line AB?
7. What is the y-intercept of the line which passes through point C and is parallel to line AB?
8. What is the equation of the line in slope-intercept form which passes through point C and is parallel to line AB?
9. Draw the line in part 8 on your graph paper.
10. Does this line look parallel to line AB? (yes or no)
11. Does the y-intercept of the graph match the value you calculated in part 7? (yes or no)
12. What is the slope of the line which passes through point C and is perpendicular to line AB?
13. What is the y-intercept of the line which passes through point C and is perpendicular to line AB?
14. What is the equation of the line which passes through point C and is perpendicular to line AB?
15. Draw the line in part 14.
16. Does this line look perpendicular to line AB? (yes or no)
17. Does the y-intercept of the graph match the value you calculated in part 13? (yes or no)
Question #2: 10 points
Let the equation of a line in general form (Ax + By = C) be : 3x + 5y = 15.
1. Re-write the equation so that it is in slope-intercept form (p.102 of the book).
2. Re-write the equation so that it is in point-slope form (p.107 of the book). Clearly show how you got the ordered pair (x1,y1) to put in your equation.
Question #3: 15 points
Let 5x + 15 > 15. Let -2x +6 >= 12.
1. Solve the first inequality for x.
2. Write your answer in interval notation.
3. Plot your first answer on a graph.
4. Solve the second inequality for x.
5. Write your answer in interval notation.
6. Plot your second answer on a graph.
7. Write your combined answer in interval notation for the OR solution for x (p124 of book).
8. Write your combined answer in interval notation for the AND solution for x (p125 of book).
Question #4: 20 points
Altimeter data from NASA satellites have been measuring sea level relative to the center of the earth since 1992. In 1993, the average global sea level height was -12 mm on an arbitrary scale. In 2014, the average global sea level height was +55 mm on the same scale. Let x equal the number of years after 1993.
1. Write two ordered pairs to represent the information given above.
2. Find the slope of the line which passes through these two points.
3. Write the equation of the line in point-slope form.
4. Write the equation of the line in slope-intercept form.
5. Draw .
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5.1 writing linear equations in slope intercept form
- 1. CHAPTER 5: WRITING LINEAR EQUATIONS 5.1 WRITING LINEAR EQUATIONS IN SLOPE-INTERCEPT FORM BY: ANIBAL AGUILAR BARAHONA
- 2. 5.1 WRITING LINEAR EQUATIONS IN SLOPE-INTERCEPT FORM During this lesson, you will learn how to write an equation of a line using as information, the slope (m) and the y-intercept (b). To do this, you need to remember the slope-intercept form of a linear equation: 𝒚 = 𝒎𝒙 + 𝒃 You also need to remember that there are different ways to represent the slope of a line: 𝒎 = ∆𝒚 ∆𝒙 = 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒚 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒙 = 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 The y-intercept is the point in the y-axis were the graph of the line is crossing.
- 3. 5.1 WRITING LINEAR EQUATIONS IN SLOPE-INTERCEPT FORM Below, you will see the most typical examples in writing equations of a line using the slope and y-intercept. In some cases, you only need to replace the given values in the slope-intercept form of the linear equation. a. Write an equation of the line whose slope is – 2 and whose y-intercept is 5. b. Write an equatons of the line whose slope is − 2 5 and whose y-intercept is – 10 .
- 4. 5.1 WRITING LINEAR EQUATIONS IN SLOPE-INTERCEPT FORM Information can be presented in different ways in the process of writing an equation of a line. Some examples can be seen below: c. Write an equation of a line passing through 0, −4 that has a change in y of 5 and a change in x of 3. d. Write an equation of a line passing through 0, 12 that has ∆𝑥 = 4 𝑎𝑛𝑑 ∆𝑦 = −8.
- 5. HOMEWORK: PG. 276 EXERCISES 3 to 8, and 12 to 19
- 6. 5.1 WRITING LINEAR EQUATIONS IN SLOPE-INTERCEPT FORM Another way to write equations of a line is by reading information from a graph. In this particular situation, you must follow these steps: • Identify the y-intercept in the graph. • Identify the slope of the line, by finding ∆𝑥 𝑎𝑛𝑑 ∆𝑦, to determine the slope. This can be done by identifying “the triangle” in the line. Examples: Write an equation of the line shown in the graph. 1.
- 7. 5.1 WRITING LINEAR EQUATIONS IN SLOPE-INTERCEPT FORM 2. 3.
- 8. 5.1 WRITING LINEAR EQUATIONS IN SLOPE-INTERCEPT FORM 4.
- 9. HOMEWORK: PG. 276 EXERCISES 20 to 25