GRE: Graphical Representations COORDINATE GEOMETRY

The Coordinate Plane Coordinate planes are formed by two axes: the x-axis and the y-axis The point where the two axes meet is called the origin. The coordinates of the origin are (0, 0). Coordinate planes are divided into 4 quadrants.

Plotting Points in the Coordinate Plane Points in a coordinate plane are located according to their coordinates. Coordinates are written as a pair: (x-coordinate, y-coordinate) The x-coordinate indicates how far left or right the point is from the origin. The y-coordinate indicates how far up or down the point is from the origin Example – the coordinates of the point on the coordinate plane is (3, 1)

The Midpoint Formula To find the midpoint or the halfway point between two points, we use the Midpoint Formula: Example 1 – find the midpoint between (-2, 5) and (4, -1) Solution: x1 = -2, x2 = 4, y1 = 5, and y2 = -1 So, (1, 2) is the midpoint between (-2, 5) and (4, -1)

Visual Representation of Midpoint Formula Notice that the midpoint is always on the line segment that connects the two endpoints.

Another Example Example 2 – Find the midpoint between (0, 6) and (-4, -2). Solution: (-2, 2) So, (-2, 2) is the midpoint between (0, 6) and (-4, -2)

You Try Find the midpoint between (-7, 2) and (5, -8).

Homework Do all of the problems on GRE Practice Problems 1. For 3rd, 7th , and 8th periods #6 is extra credit.

Extensions of the Midpoint Formula How do we find the other endpoint of a line segment when given one endpoint and the midpoint? - (x2, y2) = (2xm – x1, 2ym – y1) where (x2, y2) is the missing endpoint. Example 1 – Find the other endpoint of a line segment if one endpoint is (-2, 7) and the midpoint is (3, -1). Solution: (x2, y2) = (2·3 - -2, 2·-1 – 7) = (6 + 2, -2 – 7) = (8, -9)

More Examples Example 2 – Find the other endpoint of a line segment if one endpoint is (-5, -6) and the midpoint is (2, 3). Solution: (x2, y2) = (2 · 2 - -5, 2 · 3 - -6) = (4 - -5, 6 - -6) = (9, 12) Example 3 – Find the other endpoint of a line segment if one endpoint is (8, 4) and the midpoint is (-1, 10) Solution: (x2, y2) = (2 · -1 – 8, 2 · 10 – 4) = (-2 – 8, 20 – 4) = (-10, 16)

You Try 1. Find the other endpoint of a line segment if one endpoint is (-11, -3) and the midpoint is (-7, 7). 2. Find the other endpoint of a line segment if one endpoint is (15, -12) and the midpoint is (-3, -5).

Homework Do all of the problems on GRE Practice Problems 2. Number 7 is extra credit for all classes.

Finding the Distance Between Two Points Draw a line segment connecting the two points. Count the number of units going horizontally from one endpoint to the other. (8) Count the number of units going vertically from endpoint to the other. (8) Square both of those numbers and add them together. (8² + 8² = 128) Find the square root of that sum. Simplify the radical if possible. (√128 = 8√2) So, the distance from (-5, -3) to (3, 5) is 8√2 units.

The Distance Formula What is the distance formula? - We can use the distance formula to find the distance between two points or to determine how long a line segment is. - Distance Formula: - (x1, y1) and (x2, y2) are the endpoints Example 1 – Find the distance between (-5, -3) and (3, 5). Solution:

More Examples Example 2 – Find the distance between (3, 7) and (-2, -6). Solution: Example 3 – Find the distance between (-1, -5) and (2, 0).

You Try Find the distance between (-4, 5) and (0, -1).

Homework Do all of the problems on GRE Practice Problems 3. For 3rd , 7th, and 8th periods #6 is extra credit.

Finding the Slope Between Two Points What is slope? - Slope is the steepness of a line - Slope is the rate at which “y” changes as “x” increases by 1 - Slope is the rise of a line divided by its run - “m” is the symbol used to represent slope Slope Formula: where (x1, y1) and (x2, y2) are points on the line

More Points About Slope If a line goes down going from left to right, then the slope of the line is negative. If a line goes up going from left to right, then the slope of the line is positive. The steeper the line, the greater the slope. The slope of horizontal lines is 0. The slope of vertical lines is undefined.

Using the Slope Formula Example 1 – Find the slope of the line that passes through (8, 2) and (-3, -4). Solution: Example 2 – Find the slope of the line that passes through (-7, 4) and (-5, -10). Which line in the examples is steeper? How do you know?

You Try Find the slope of the line that passes through (5, -3) and (2, 3). Find the slope of the line that passes through (-11, -9) and (11, -11). Which of the lines is steeper?

Homework Do all of the problems on GRE Practice Problems 4.

Finding the Slope of a Line When finding the slope of a line, Find two points through which the line passes. Plug those points into the slope formula. Example 1 – The line passes through (-6, 5) and (0, -2) -2-5/0 - -6 = -8/6 = -4/3

You Try 1. Find the slope of the line in the coordinate plane on the right.

Homework Do all of the problems on GRE Practice Problems 5. For 3rd, 7th, and 8th periods #5 is extra credit.

Forms of Linear Equations Lines in coordinate planes can be represented by equations There are 3 basic forms of linear equations: 1. slope-intercept form 2. point-slope form 3. standard form Each form has its own unique characteristics

Slope-Intercept Form y = mx + b This form explicitly gives the slope (m) of the line and the y-intercept (b) The y-intercept is the point where the line crosses the y-axis (0, b) x and y are general variables and are not replaced by numbers However, m and b are replaced by numbers

Examples of Slope-Intercept Form Example 1: y = -4x + 5 m = -4 and b = 5, so the slope of the line is -4 and the point where the line crosses the y-axis is (0, 5) Example 2: y = 7x m = 7 and b = 0, so the slope of the line is 7 and the point where the line crosses the y-axis is (0, 0)

We Try What is the slope and y-intercept of the line whose equation is y = 9x + 4? 2. What is the slope and y-intercept of the line whose equation is y = -6x?

Point-Slope Form y – y1 = m(x – x1) This form explicitly gives the slope (m) of the line and a point through which the line passes (x1, y1). x and y are general variables and are not replaced by numbers However, m, x1, and y1 are replaced by numbers

Examples of Point-Slope Form Example 1: y – 3 = 5(x – 1) m = 5, x1 = 1, and y1 = 3 So, the slope of the line is 5 and it passes through the point (1, 3) Example 2: y + 8 = -2(x + 4) m = -2, x1 = -4, and y1 = -8 So, the slope is -2 and the line passes through the point (-4, -8) *Note: the coordinates of the point are always the opposite of the numbers in the equation.

You Try y – 10 = -6(x + 4) What is the slope? Through what point does the line pass? y + 3 = -(x – 2)

Standard Form ax + by = c This form does not explicitly give anything x and y are general variables and are not replaced by numbers However, a, b, and c are replaced by numbers where a and b are coefficients and c is a constant m = -a/b  the slope of the line in standard form is the opposite of a divided by b

Examples of Standard Form Example 1: 8x – 4y = 12 a = 8, b = -4, and c = 12 m = -8/-4 = 2 So, the slope of the line is 2. Example 2: 7x + y = 11 a = 7, b = 1, and c = 11 m = -7/1 = -7 So, the slope of the line is -7.

You Try Find the slope of the line whose equation is -4x – 2y = 16

Form of Linear Equation Key Info In Equation Slope-Intercept y = mx + b Slope (m), y-intercept (0, b) Point-Slope y – y1 = m(x – x1) Slope (m), point through which line passes (x1, y1) Standard Form ax + by = c Slope (m) = -a/b

Special Linear Equations y = constant Horizontal line slope = 0 Example: y = 4 A horizontal line that goes through (0, 4) x = constant Vertical line slope is undefined Example: x = 4 A vertical line that goes through (4, 0)

You Try Some More… Identify the form and find the key info from the following equations: y – 2 = -(x + 5) 4x – 3y = 15 y = -7x – 2 -x + 4y = -10 y + 7 = 3(x – 1)

Homework Complete all of the problems on GRE Practice Problems 6.

Parallel Lines Do not intersect Have the same slope

Examples of Parallel Lines Example 1: equation of line a is y = 2x + 4 equation of line b is y = 2x – 3 Lines a and b are parallel because they have the same slope: 2 Example 2: equation of line c is 3x – y = 4 equation of line d is y – 3 = 3(x + 6) Lines c and d are parallel because they have the same slope: 3

You Try Determine whether or not the following pairs of lines are parallel: 6x + 9y = 18 y = 2/3 x – 4 y – 8 = -2/5 (x + 6) -4x – 10 y = 20

Perpendicular Lines Intersect at a right angle Slopes are opposite reciprocals

Examples of Perpendicular Lines Example 1: What is the slope of line b if it is perpendicular to line a whose equation is y = -3x + 6? ma = -3  mb = -(1/-3) = 1/3 Example 2: What is the slope of line d if it is perpendicular to line c whose equation is 10x – 5y = 15? mc = -10/-5 = 2  md = -(1/2) = -1/2

You Try If lines t and u are perpendicular and the equation of line u is 4x – 8y = 12, then what is the slope of line t? 2. If lines e and f are perpendicular and the equation of line f is y – 1 = -3/7(x + 4), then what is the slope of line e?

Think About It… If a line is horizontal, then what would be the slope of the line that is perpendicular to it? If a line is vertical, then what would be the slope of the line that is perpendicular to it?

Homework Complete all of the problems on GRE Practice Problems 7.

Converting From Point-Slope to Slope-Intercept Point-Slope Form: y – y1 = m(x – x1) Slope-Intercept Form: y = mx + b What do these two equations have in common? What is different about these two equations? We must get “y” by itself on the left side of the equal sign.

Example of Converting From Point-Slope to Slope-Intercept Since point-slope form and slope-intercept form both have “x”, “y”, and “m”, we just need to rearrange the variables using some algebraic properties. Example 1: Convert y – 3 = 4(x + 5) to slope-intercept form. Solution: Step 1 y – 3 = 4x + 20 Distributive Property Step 2 y – 3 = 4x + 20 +3 + 3 Inverse Operations y = 4x + 23

More Examples of Converting From Point-Slope to Slope-Intercept Example 2: Convert y + 6 = -2(x – 8) to slope-intercept form. Solution: Step 1 y + 6 = -2x + 16 Dist Prop Step 2 y + 6 = -2x + 16 - 6 - 6 Inverse Op y = -2x + 10

More Examples of Converting From Point-Slope to Slope-Intercept Example 3: Convert y + 8 = -3/4(x – 12) to slope-intercept form. Solution: Step 1 y + 8 = -3/4 x + 9 Dist. Prop Step 2 y + 8 = -3/4 x + 9 - 8 - 8 Inverse Op. y = -3/4 x + 1

You Try Convert y + 7 = -4(x – 3) to slope-intercept form.

Homework Do all of the problems on GRE Practice Problems 9.

Converting From Standard Form to Slope-Intercept Form Standard Form: ax + by = c Slope-Intercept Form: y = mx + b To convert from standard form to slope-intercept form, we must get “y” by itself on the left side of the equal sign by using inverse operations.

Example of Converting from Standard Form to Slope-Intercept Example 1: Convert 8x – 4y = 16 to slope-intercept form. Solution: Step 1 8x – 4y = 16 -8x - 8x -4y = 16 – 8x Step 2 -4y/-4 = 16/-4 – 8x/-4 y = -4 + 2x Step 3 y = 2x – 4

More Examples of Converting from Standard Form to Slope-Intercept Example 2: Convert -12x – 3y = -24 to slope-intercept form. Solution: Step 1 -12x – 3y = -24 +12x + 12x -3y = -24 + 12x Step 2 -3y/-3 = -24/-3 + 12x/-3 y = 8 + -4x Step 3 y = -4x + 8

Recap on Converting from Standard Form to Slope-Intercept Step 1 – add/subtract the “x” term from both sides of equation Step 2 – divide all terms by the coefficient of y Step 3 – rearrange the terms on the right side of the equation so that the “x” term is first

You Try Convert 10x + 2y = 20 to slope-intercept form.

Graphing Linear Equations In order to graph linear equations, we must either know (1) two points that the line passes through or (2) the slope of the line and a point that the line passes through. For example, if we want to graph the equation y = 3x + 4, then we know the slope is 3 and the y-intercept is (0, 4). So, we’ll plot the point (0, 4) and find another point using the slope.

More Notes on Graphing Linear Equations If the slope is positive, then go up the amount in the numerator and right the amount in the denominator. If the slope is negative, then go down the amount in the numerator and right the amount in the denominator.

y = 3x + 4 y-intercept: (0, 4) m = 3/1

You Try Graph the following equations: y = -3x + 7 y = ½ x – 8

Homework Do all of the problems on GRE Practice Problems 11. Study for quiz.

Writing Equations of Lines When Given Two Points There are two key pieces of information that we need when writing an equation of a line: 1. The slope of the line 2. A point through which the line passes

Example of Writing An Equation of a Line When Given Two Points Example 1: Write an equation of the line that passes through points (4, 0) and (6, -4). Solution: Step 1 – Find the slope of the line m = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 = −4 − 0 6 − 4 = −4 2 = -2 Step 2 – Plug slope and either point into point-slope form y – y1 = m(x – x1) y + 4 = -2(x – 6) Step 3 – Put in slope-intercept form y + 4 = -2x + 12 -4 - 4 y = -2x + 8

Another Example of Writing An Equation of a Line When Given Two Points Example 2: Write an equation of the line that passes through points (-1, -5) and (2, 4). Solution: m = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 = 4−(−5) 2−(−1) = 9 3 = 3 y – y1 = m(x – x1) y + 5 = 3(x + 1) y + 5 = 3x + 3 - 5 - 5 y = 3x - 2

You Try Write an equation of the line that passes through points (4, -1) and (9, 14). 2. Write an equation of the line that passes through points (10, -3) and (9, -8).

Homework Complete all of the problems on GRE Practice Problems 12.

Writing Equations of Linear Graphs Again, the two key pieces of information that we need in order to write an equation of a line are (1) the slope of the line and (2) a point through which the line passes.

Example of Writing An Equation of A Linear Graph Example 1: Write an equation of the line. Solution: Step 1 – find two points that the line passes though (-3, -2) and (0, -4) Step 2 – find slope of line using the two points m = −4−(−2) 0−(−3) = −2 3 Step 3 – plug either point and slope into point-slope form y + 2 = − 2 3 (x + 3) Step 4 – put in slope-intercept form y + 2 = − 2 3 x – 2 - 2 - 2 y = − 𝟐 𝟑 x – 4

Another Example of Writing An Equation of A Linear Graph The two points that the line passes through are (2, -2) and (-2, -4). m = −4−(−2) −2−2 = −2 −4 = 1 2 y + 4 = 1 2 (x + 2) y + 4 = 1 2 x + 1 - 4 - 4 y = 𝟏 𝟐 x – 3

Another Example Write an equation of the line.

Another Example Write an equation of the line.