Warm-up What is the distance formula between two points? What is the midpoint formula of a segment? Given P(-5,9) and Q (-8,-7) Find the distance d(P,Q) … and the midpoint of the segment PQ Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Warm-up Answers Distance formula = Midpoint = d(P,Q) = M (-13/2, 1) Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Digital Lesson 2.2 Graphs of Equations

Example: Sketch the graph of y = –2x + 3. 8 4 x 4 4 8 –4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example continued

Example: Sketch Graph (Quadratic Function) Example: Sketch the graph of y = (x – 1)2. y x y (x, y) –2 9 (–2, 9) –1 4 (–1, 4) 1 (0, 1) (1, 0) 2 (2, 1) 3 (3, 4) (4, 9) 8 6 2 x –2 2 4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Sketch Graph (Quadratic Function)

Example: Sketch Graph (Absolute Value Function) Example: Sketch the graph of y = | x | + 1. y x y (x, y) –2 3 (–2, 3) –1 2 (–1, 2) 1 (0, 1) (1, 2) (2, 3) 4 2 x –2 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Sketch Graph (Absolute Value Function)

Definition of Intercepts The points at which the graph intersects the x- or y-axis are called intercepts. If (x, 0) satisfies an equation, then the point (x, 0) is called an x-intercept of the graph of the equation. If (0, y) satisfies an equation, then the point (0, y) is called a y-intercept of the graph of the equation. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition of Intercepts

Example: Find Intercepts Example: Find the x- and y-intercepts of the graph of y = x2 + 4x – 5. To find the x-intercepts, let y = 0 and solve for x. 0 = x2 + 4x – 5 Substitute 0 for y. 0 = (x – 1)(x + 5) Factor. x – 1 = 0 x + 5 = 0 Set each factor equal to 0. x = 1 x = –5 Solve for x. So, the x-intercepts are (1, 0) and (–5, 0). To find the y-intercept, let x = 0 and solve for y. y = 02 + 4(0) – 5 = –5 So, the y-intercept is (0, –5). Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Find Intercepts

Symmetries of Graphs of Equations in x and y Symmetric with respect to the y-axis: Substitution of –x for x leads to the same equation (This is also called an even function) Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Symmetric with respect to the x-axis: Substitution of –y leads to the same equation Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Symmetric with respect to the origin: Simultaneous substitution of –x for x and –y for y leads to the same equation (This is also called an Odd function) Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Circles Standard Equation of a circle with Center (h, k) and radius r (x-h)2 + (y-k)2 = r2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Semicircles Top half (solve for y, + √ ) Bottom half (solve for y, - √ ) Right half (solve for x, + √ ) Left half (solve for x, - √ ) Copyright © by Houghton Mifflin Company, Inc. All rights reserved.