PC CHAPTER 2 GUIDED NOTES.pdf

Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions Notes/Examples • Intercepts are points where a graph intersects the ____...

Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions Notes/Examples • Intercepts are points where a graph intersects the _____ or _____- _________. x- and yIntercepts • An x-intercept occurs where ___________. x-intercept x-intercept • A y-intercept occurs where ___________. • A function can have 0+ x-intercepts, y-intercept but at most one y-intercept. Examples Directions: Identify the x- and y-intercepts of the functions graphed below. 1. 2. 3. x-intercept(s): x-intercept(s): x-intercept(s): y-intercept: y-intercept: y-intercept: 4. 5. 6. x-intercept(s): x-intercept(s): x-intercept(s): y-intercept: y-intercept: y-intercept: Zeros Find Zeros & the y-intercept Algebraically • To find the zeros, set ____________________ and solve for ______. • To find the y-intercept, find ______________. © Gina Wilson (All Things Algebra®, LLC), 2017 Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions Notes/Examples • Critical points are points where the graph maximum changes its __________________ or ______________. CRITICAL POINTS • Extrema are points where the graph changes _____________________. The function has a minimum or maximum value (relative or absolute) at these points. point of inflection • A point of inflection is where the graph changes its ______________, but not its direction. minimum A point with the greatest value compared to points around it. (also called a local maximum) Relative & Absolute EXTREMA absolute maximum relative maximum The point with the greatest value over the entire domain of the function. A point with the least value compared to points around it. (also called a local minimum) relative minimum The point with the least value over the entire domain of the function. EXAMPLES absolute minimum Directions: Give the coordinates and classify the extrema for the graph of each function. Use your graphing calculator to approximate if needed. 2 1. f ( x) = 2 x − 16 x + 27 4 2 2. f ( x) = − x + 4 x − 2 x − 4 © Gina Wilson (All Things Algebra®, LLC), 2017 3 2 5 3. f ( x) = − x − 5 x − 3 x + 4 3 4. f ( x) = x − 3 x + 3 Moving from left to right, a function may increase, decrease, or remain constant on certain intervals. INCREASING & DECREASING Behavior EXAMPLES For example, the function to the right is • increasing on the interval ____________. • constant on the interval ____________. • decreasing on the interval ____________. Directions: Give the intervals on which each function increases or decreases. Use your graphing calculator to approximate if necessary. 5. 6. 7. Increasing Interval(s): Decreasing Interval(s): 3 2 8. f ( x) = − x + 4 x − 2 Increasing Interval(s): Decreasing Interval(s): Increasing Interval(s): Decreasing Interval(s): 4 3 2 9. f ( x) = x + x − 4 x + 1 Increasing Interval(s): Decreasing Interval(s): Increasing Interval(s): Decreasing Interval(s): 5 10. f ( x) = x − 2 x Increasing Interval(s): 3 Decreasing Interval(s): © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________ Unit 2: Functions & Their Graphs Date: ________________________ Per: ______ Homework 2: Intercepts, Zeros, Extrema ** This is a 2-page document! ** Directions: Identify the x- and y-intercepts of the functions graphed below. 1. 2. 3. x-intercept(s): x-intercept(s): x-intercept(s): y-intercept: y-intercept: y-intercept: 4. 5. 6. x-intercept(s): x-intercept(s): x-intercept(s): y-intercept: y-intercept: y-intercept: Directions: Find the zero(s) and y-intercept of each function algebraically. 8. f ( x) = x 3 − 10 x 2 + 24 x 7. f ( x) = 2 x − 5 − 1 zero(s): y-intercept: 4 2 9. f ( x) = 4 x − 17 x + 4 zero(s): zero(s): 10. f ( x) = y-intercept: zero(s): y-intercept: 1 x−3 2 y-intercept: © Gina Wilson (All Things Algebra®, LLC), 2017 Name: _________________________________________________ Pre-Calculus Date: ______________________________Per: ________ Unit 2: Functions & Their Graphs Quiz 2-1: Characteristics of Functions (Part 1) Directions: Give the domain and range of each relation using set notation. Then, indicate whether the relation shown by the graph is a function. 1. 2. 3. D = ________________________ D = ________________________ D = ________________________ R = ________________________ R = ________________________ R = ________________________ Function? _________ Function? _________ Function? _________ Directions: Write each equation explicitly in terms of x. Write your answer in the box, then indicate whether the equation is a function. 1 2 4. 6 x − 3 y = 21 5. y − x + 2 x = −5 6. 8 x 2 + 4 y 2 = 12 3 Function? __________ Function? __________ Function? __________ Directions: Given f(x) = x3 + 2x and g(x) = -x2 – 5x + 12, find each function value. Write your answers to the right. 7. f (−3) 8. g (9) 7. __________________ 8. __________________ 9. __________________ 9. g ( −5a) 10. f (4c − 1) 10. __________________ © Gina Wilson (All Things Algebra®, LLC), 2017 Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions Notes/Examples • A graph is said to be ______________________________ if there are no ______________ or _________________ in Continuity the graph. • If there are breaks or holes in the graphs, it is considered to be ______________________________. Types of Discontinuity a When f(x) increases or decreases infinitely as x approaches x = a from the left or the right. Examples a a When the f(x) jumps from one point to another at x = a. When f(x) is continuous everywhere else except for a hole at x = a. Directions: Determine if the function shown on the graph is continuous. If not, identify the type and location of discontinuity. 1.\ 2. 3. 4. 5. 6. © Gina Wilson (All Things Algebra®, LLC), 2017 End Behavior As x moves towards positive infinity and negative infinity, end behavior describes what is happening to the value of f(x). Describe the end behavior of the graph to the right: As x → ∞ , f ( x) → ________ As x → −∞ , f ( x) → ________ Examples Directions: Describe the end behavior of each graph. 7. 8. 9. As x moves towards positive infinity and negative infinity, it is possible for f(x) to be approaching a certain value. Graphs Approaching a Certain Value Describe the end behavior of the graph to the right: As x → ∞ , f ( x) → ________ As x → −∞ , f ( x) → ________ Examples Directions: Describe the end behavior of each graph. 10. 11. 12. 13. 14. 15. © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________ Unit 2: Functions & Their Graphs Date: ________________________ Per: ______ Homework 3: Continuity & End Behavior ** This is a 2-page document! ** Directions: Determine if the function shown on the graph is continuous. If not, identify the type and location of discontinuity. 1. 2. 3. 4. 5. 6. 7. 8. 9. © Gina Wilson (All Things Algebra®, LLC), 2017 Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions TYPES OF SYMMETRY Notes/Examples The graph can be folded along a line so that the two halves match. The graph remains unchanged when rotated 180° about a point. y y x y x x The _____-___________ and _____-___________ are common lines of symmetry. The _______________ is a common point of symmetry. The following graphical and algebraic tests can be used to determine if the graph of a relation is symmetric to the x-axis, y-axis, and/or origin. TESTS FOR SYMMETRY origin y-axis x-axis Graphical Test Algebraic Test For every point (x, y) on the Replacing ______ with ______ graph, the point _________ is results in equivalent equations. also on the graph. For every point (x, y) on the Replacing ______ with ______ graph, the point _________ is results in equivalent equations. also on the graph. For every point (x, y) on the Replacing ______ with ______ graph, the point _________ is AND ______ with ______ results also on the graph. in equivalent equations. Directions: Use the graph to determine if the relation is symmetrical to the Examples x-axis, y-axis, and/or origin. Confirm your answer algebraically. 1. y = x 4 − 4 x 2



x-axis y-axis origin none 2. xy = 6



x-axis y-axis origin none © Gina Wilson (All Things Algebra®, LLC), 2017 3. 4( x + 3)2 + y 2 = 16



5. y =



4. y 2 − x 2 = 1



x-axis y-axis origin none 3 x −1 x-axis y-axis origin none 6. y = − x − 2



x-axis y-axis origin none x-axis y-axis origin none Algebraic Check: A function is __________ if it is symmetric with EVEN AND ODD respect to the _____-__________. Functions A function is __________ if it is symmetric with Algebraic Check: respect to the ________________. Examples Directions: Determine algebraically if the function is even, odd, or neither. If even or odd, describe the symmetry. 3 2 7. f ( x) = x − x 8. f ( x) = − x + 6 9. 11. f ( x) = x 3 − x 2 − 2 x f ( x) = −8 x x2 − 4 10. f ( x) = 12. f ( x) = x + 1 © Gina Wilson (All Things Algebra®, LLC), 2017