Section 2.2 Intercepts & Symmetry Intercepts Week 2 Handout MAC 1105 Professor Niraj Wagh J The x-intercept(s) of an equation on a graph is the point on the graph where y = 0. è So to find the x-intercept, let y = 0. The y-intercept(s) of an equation on a graph is the point on the graph where x = 0. è So to find the y-intercept, let x = 0. Symmetry X-Axis Symmetry A graph is said to be symmetric about the x-axis if, for every point (x, y) on the graph, the point (x, -y) is also on the graph. è To test if the graph of an equation is symmetric about the x-axis, replace y with negative y and simplify. If the same equation exists then it is symmetric about the x-axis. Y-Axis Symmetry A graph is said to be symmetric about the y-axis if, for every point (x, y) on the graph, the point (-x, y) is also on the graph. è To test if the graph of an equation is symmetric about the y-axis, replace x with negative x and simplify. If the same equation exists then it is symmetric about the y-axis. N. Wagh 1
Origin Symmetry A graph is said to be symmetric about the origin if, for every point (x, y) on the graph, the point (-x, -y) is also on the graph. è To test if the graph of an equation is symmetric about the origin, replace both x with negative x and y with negative y and simplify. If the same equation exists then it is symmetric about the origin. 2.2 Example (a) (b) (c) Decide if the points (0, 0), (7, 4), or (1, 5) are solutions to the equation. Find the x and y-intercepts. Decide if the equation exhibits x, y, or origin symmetry. 1. y $ = x + 9 N. Wagh 2
Section 2.3 Lines Slope Let P 1 = (x 1, y 1 ) and P 2 = (x 2, y 2 ) be two distinct points. The slope m containing P 1 and P 2 is: m = rise run m = y 2 y 1 OR m = y 1 y 2 x 2 x 1 x 1 x 2 N. Wagh 3
Notes: If x 1 = x 2, the slope of the line will be undefined since we cannot divide by zero. è The resulting line is vertical. (UNDEFINED SLOPE, Form: X = some #) Mr. Penguin: AHH!!! HELP ME!! L If y 1 = y 2, the slope of the line will be 0 and zero divided by anything will be zero. è The resulting line is horizontal. (NO SLOPE, Form: Y = some #) Mr. Penguin: Umm.. I m not going anywhere, this is boring! N. Wagh 4
Positive Slope Mr. Penguin: I m tired. Negative Slope Mr. Penguin: Weee!!!! Credits for images: themathroundtable.wikispaces.com/speedo's+page http://regentsprep.org/regents/math/geometry/gcg1/eqlines.htm Forms of Equations of Lines Slope-Intercept Form of an Equation of a Line An equation with slope m and y-intercept b is y = mx + b Point-Slope Form of an Equation of a Line An equation of a non-vertical line with slope m that contains point (x 1, y 1 ) is General Form of an Equation of a Line y y 1 = m(x x 1 ) Ax + By = C Where A, B, and C are real numbers and A and B are not both 0. N. Wagh 5
Parallel Lines Two nonvertical lines are said to parallel with one another if and only if they have the same slope and different y-intercepts. è Have the same slope. è Have different y-intercepts. NOTE: If the lines had the same y-intercept they would be the same line. :P Perpendicular Lines Two nonvertical lines are perpendicular (when two lines intersect at a right angle [90 0 ]) if and only if the product of their slopes is -1. NOTE: You can also say that two nonvertical lines are perpendicular if the slopes are negative reciprocals of one another. è For example, if one line had a slope of 5 and the other line had a slope of -1/5 then they are perpendicular lines. o This is because 5*-1/5 = -1. J 2.3 Examples Find an equation for the line with the given properties. Express your answers using slope-intercept form. 1. Containing the points (-3, 4) and (2, 5) N. Wagh 6
2. Parallel and Perpendicular to the line y = 2x 3; containing the point (1, -2) 3. Gloria loves going to concerts. Her favorite artist is Kendrick Lamar. When she heard Kendrick was coming to town she knew she had to go! She wants to meet Kendrick so she buys a meet and greet pass for 499 dollars plus 6 dollars for each type of merchandise she buys. Write a linear equation that relates the cost C, in dollars, of buying a ticket to the number x of merchandise she buys. What is the total cost of the concert if Gloria gets 5 types of merchandise? 10 types of merchandise? N. Wagh 7
PRACTICE. PRACTICE. PRACTICE. 1. y 5x $ = 4 (a) (b) (c) Decide if the points (0, 4), (1, 3), or (3, 3) are solutions to the equation. Find the x and y-intercepts. Decide if the equation exhibits x, y, or origin symmetry. 2. Find an equation that is perpendicular to y = -4x + 10 that passes through the point (7, 2). 3. Find an equation that goes through the points (2, 5) and (10, 1). We are done! Now work on HW 2.2, 2.3. If you have any questions, please let me know! N. Wagh 8
Section 2.4 Circles A circle is a set of points in the xy-plane that are a fixed distance r from a fixed point (h, k). The fixed distance r is called the radius, and the fixed point (h, k) is called the center of the circle. Forms The standard form of an equation of a circle with radius r and center (h, k) is (x h) 2 + (y k) 2 = r 2 The general form of the equation of a circle is x 2 + y 2 + ax + by + c = 0 Hey I m a circle too! N. Wagh 9
Completing The Square Steps Use this technique to factor the general form. è Recall the general form of the equation of a circle is x 2 + y 2 + ax + by + c = 0 1. Make sure that the coefficients for x 2 and y 2 are 1. 2. Rearrange so that you get the form: (x 2 + ax)+ (y 2 + by) = c 3. Take half of the a and b term, these are the coefficients of the x and y terms respectively. 4. Square those numbers and add it each respective number within the parentheses. 5. Now add the numbers to the other side of the equation. 6. Factor each and simplify. 2.4 Examples Write the standard and general form of the equation of each circle of radius r and center (h, k). Then graph the circle. 1. r = 3; (h, k) = (0, 0) N. Wagh 10
First, find the center (h, k) and radius of the circle. Second, graph the circle. And lastly, find the intercepts, if any. 2. 3x 2 + 3y 2 24x + 6y 24 = 0 N. Wagh 11
PRACTICE. PRACTICE. PRACTICE. 3. x 2 + y 2 + 4x + 2y 20 = 0 4. x 2 + y 2 6x 8y 11= 0 5. x 2 + y 2 4x 6y 23 = 0 6. x 2 + y 2 3x + 4y + 4 = 0 7. 5x 2 + 5y 2 30x +10y 75 = 0 Yay! Now work on HW 2.4. If you have any questions, please let me know! N. Wagh 12