GSE Algebra 1 Name Date Block. Unit 3b Remediation Ticket

Transcription

1 Unit 3b Remediation Ticket Question: Which function increases faster, f(x) or g(x)? f(x) = 5x + 8; two points from g(x): (-2, 4) and (3, 10) Answer: In order to compare the rate of change (roc), you must find the slope. The slope of f(x) is taken from slope-intercept form y = mx + b, where m is the slope. So the slope of f(x) is 5 or 5 in which you 1 go up 5 and right 1. The slope of g(x) can be found by the slope formula m = y 2 y 1 = 10 4 = 6 = 1.2. Since the slope of x 2 x 1 3 ( 2) 5 f(x) is greater, the function f(x) increases faster. Example 0- You Try!! Identify which function has a greater rate of change. a. f(x) = 3x 2 and g(x): x g(x) b. f(x) = 3x 2 and g(x): Question: What is a maximum or minimum value? Answer: The y-value of the vertex. Example 1- You Try!! Identify whether the function has a maximum or minimum. State the value. a. b.

2 Question: How do I know if a quadratic function has a maximum or minimum if I m only given an equation and not a graph? Standard: y = ax 2 + bx + c Vertex: y = a(x h) 2 + k Answer: Look at the a value (which is the leading coefficient-the first thing that you see). If a is negative, the graph opens downward, meaning the vertex is at the top, meaning it has a maximum. If a is positive, the opens up, meaning the vertex is at the bottom, meaning it has a minimum. a > 0 a < 0 a is positive, a is negative graph opens up graph opens down Vertex is at the bottom - MINIMUM Vertex is at the top - MAXIMUM Example 2 - YOU TRY!!: Identify whether the following quadratic functions have a maximum or minimum. a. f(x) = 2(x 3) 2 1 b. g(x) = 5x c. h(x) = 1 (x 7)2 2 Question: How do I actually find the maximum or minimum value from an equation if I m not given the graph? Answer: What you do depends on what form the quadratic is in. Remember, the max or min is just the y-value of the vertex. Therefore, you need to find the vertex. Your answer will be the y-value. Follow-up question: Okay, what if the equation is in vertex form How do I find the vertex in order to identify the y-value? Answer: Vertex form is y = a(x h) 2 + k. The vertex is just (h, k).

3 Example 3: State the vertex of the quadratic function. Identify the max or min. y = 2(x 4) The vertex is (4, 3), so the MIN is 3 f(x) = 1 7 (x + 1)2 4 The vertex is (-1, -4), so the MIN is -4 g(x) = 3(x + 5) 2 The vertex is (-5, 0), so the MAX is 0 y = x 2 6 The vertex is (0, -6) so the MIN is -6 Follow-up question: Okay, what if the equation is in standard form How do I find the vertex in order to identify the y-value? Answer: Standard form is y = ax 2 + bx + c. To find the vertex, just use b 2a. Example 4: State the vertex of the quadratic function. Identify the max or min. f(x) = 2x 2 4x + 6 b ( 4) 4 1 (-1,?) Plug in -1 for x! 2a 2( 2) 4 f( 1) = 2( 1) 2 4( 1) + 6 = 2(1) = 8 So the vertex is (-1, 8) which means the MAX value is 8 Question: How do I find the y-intercept from an equation? Answer: To find the y-intercept, just plug in 0 for x and solve. Example 5: Where does the function cross the y-axis? f(x) = 1 (x + 8)2 4 f(0) = 1 (0 + 8)2 4 = 1 4 (8)2 = 1 4 (64) = 16 So the y-intercept is (0, 16)

4 Question: How do I find the x-intercept (also called, zeros, roots, or solutions) from an equation? Answer: To find the x-intercept, just plug in 0 for y and solve. Example 6: For what values of x does the graph of the following function cross the x-axis? f(x) = 4x x + 16 (Remember, f(x) is the same as y, so we re going to plug in 0 for f(x)). 0 = 4x x + 16 Since we have a quadratic, there are several ways to solve for x. We can use the quadratic formula, complete the square, or solve by factoring. We can t use the square root method because we don t only have x 2 in the problem. Let s solve by factoring 0 = 4(x 2 + 5x + 4) 0 = 4(x + 4)(x + 1) So 0 = 4 0 = x = x = x 1 = x So the graph crosses the x-axis at -4 and -1. Question: How do I write the vertex form a quadratic function from a graph? Answer: The vertex form of a quadratic function is y = a(x h) 2 + k. Therefore, to write the equation, all I need is a, h, and k. Follow-up question: How do I find a? Answer: From the vertex, move 1 unit left or right. However many units you move up or down to touch the graph is your a value. If you move up, a is positive. If you move down, a is negative. Follow-up question: How do I find h and k? Answer: (h, k) is your vertex! Example 7: Write the vertex form of the given parabola. a. b.

5 Characteristics of a Quadratic Function BUT WHAT DOES THAT ALL MEAN?!? 1. Identify the following characteristics of the quadratic function below. a. Extrema: Minimum at (-3, -8) b. Domain: All real numbers; < x < c. Range: 8 y < d. End Behavior: x, y ; x, y e. Zeros: x = 5 and x = 1 or ( 5,0)and ( 1,0) f. Interval(s) of Increase: 3 < x < g. Interval(s) of Decrease: < x < 3 h. Positive Interval: < x < 5 or 1 < x < i. Negative Interval: 5 < x < 1 a. Extrema: Maximum (open down) or minimum (open up) at the vertex b. Domain: Left and right values of x c. Range: Up and down values of y (start with the vertex and look where graph goes) d. End Behavior: As you move left ( ), does graph go up or down? As you move right ( ), does graph go up or down? e. Zeros: Where does graph touch the x-axis? Where does y=0? f. Interval(s) of Increase: x values going to right; when is graph going up? g. Interval(s) of Decrease: x values going to right; when is graph going down? h. Positive Interval: x values in which graph is above x-axis i. Negative Interval: x values in which graph is below x-axis Now you try! a. Extrema: b. Domain: c. Range: d. End Behavior: e. Zeros f. Interval(s) of Increase: g. Interval(s) of Decrease: h. Positive Interval: i. Negative Interval:

6 Practice!! All problems are from the Georgia Coach TM, Algebra 1, GSE Edition, Assessments booklet, distributed by Triumph Learning TM! Take this seriously! 1. Which of the following equations represents a parabola that reaches its maximum value at (2, 10)? A. y = (x + 2) B. y = (x 2) 2 10 C. y = (x + 2) D. y = (x 2) Quincy is using function notation to describe quadratic functions. Which represents the quadratic function that exhibits the following key features? axis of symmetry: x = -3 leading coefficient: 4 y-intercept: -7 A. f(x) = 4x 2 + 6x 56 B. f(x) = 4x 2 6x + 56 C. f(x) = 4x x 7 D. f(x) = 4x 2 24x Which of the following functions has a graph that shows a parabola that opens downward? A. f(x) = 5(x 2) B. h(x) = (x + 6) 2 7 C. m(x) = 1 2 (x 1) D. p(x) = 2(x + 3) Which represents the solution set of the following equation? 5x 2 + 5x = 1 A. x = 0, 2 B. x = 1 + 5, C. x = 1, D. x = 1 + 5, For what values of x will the graph of the following function cross the x-axis? f(x) = 2x x + 30 A. { 5, 3} B. {3, 5} C. { 5, 3} D. { 3, 5}

7 6. Which describes all of the x- and y-intercepts of the function below? A. ( 16, 0), (0, 8), and (16, 0) B. (0, 16), (0, 16), and (8, 0) C. ( 8, 0) and (0, 16) D. (0, 2)and (0, 2) f(x) = 1 (x + 8)2 4 Constructed Response 7. Find the maximum value of the expression x x 10. Include a description of the method used to find the solution and include the value of x for which the maximum is found. Show all work. 8. Find the vertex of the quadratic function y = x 2 8x 5 by completing the square. Show all work. Fill in the blank. 9. Write the standard form of a quadratic equation 10. Write the vertex form of a quadratic equation 11. When a quadratic equation is in vertex form, you can easily tell what the is. 12. When a quadratic equation is in standard form, there are two ways to find the vertex. One way is using b. The other way is. 2a 13. To find the x-intercept of any function, plug in for. 14. To find the y-intercept of any function, plug in for. 15. If given a quadratic equation, I can determine if the parabola opens up or down based on the value. 16. If I m asked to find the min or max value, I should be thinking about the. 17. If is positive, the parabola opens. 18. If is negative, the parabola opens. 19. If I am asked to find the zeros of the function, that means I am looking for the intercept(s). If I have a calculator, the quickest way to do this is to go to the, type in the function, and look for the x value(s) when y is. BE CAREFUL! THE FUNCTION MUST FIRST BE SET EQUAL TO 0! (You can also use the calculator in this same way to find the y-intercept, but you ll be looking for the point where x is 0).

8 Multiple Choice. Show all work or write an explanation as to how you got your answer! 20. Which of the following equations represents a parabola that reaches its minimum value at (-1, 5)? A. y = (x + 1) C. y = (x + 1) B. y = (x 1) D. y = (x 1) Find the zeros of the following quadratic function. 5x 2 6x = 1 A. x = 1, x = 5 C. x = -1, x = 5 B. x = 1 5, x = 1 D. x = -1, x = For what values of x will the graph of the following function cross the x-axis? f(x) = 3x 2 12x + 9 A. {-3, 1} C. {3, -1} B. {-3, -1} D. {3, 1} 23. Which of the following represents a parabola with a maximum value? 24. A. f(x) = 3 4 (x 2)2 + 1 C. g(x) = x B. h(x) = 2 x + 5 D. k(x) = 5(x 1) 2 9 Constructed Response. Show all work. 25. Find the minimum value of the expression x 2 4x 5. Include a description of the method used to find the solution. 26. Find the vertex of the quadratic function f(x) = x 2 12x 4 by completing the square.