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# Ch 7 tutoring notes quadratics

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### Ch 7 tutoring notes quadratics

1. 1. Quadratics Graphs of quadratic functions in the form Transformations of the graph of the parabola f(x) = a(x-h) ² + k vertex form Quadratic Function in Standard Form:
2. 2. Transformations of f(x) = a(x-h) ² + k <ul><li>EX : </li></ul><ul><li>a  If a gets closer to 0, then the graph widens. If a gets farther from 0, then the graph narrows. If a is negative, the graph reflects across the x-axis (downward). </li></ul>
3. 3. Transformations of f(x) = a(x-h) ² + k <ul><li>h  If there is a number after x inside the function, then the graph is shifted left or right according to the opposite sign . </li></ul>
4. 4. Transformations of f(x) = a(x-h) ² + k <ul><li>k  If there is a number outside the function, then the graph shifts up or down according to the same sign. x² + 14 moves up 14 </li></ul>
5. 5. Transformations of f(x) = a(x-h) ² + k <ul><li>Write the equation for each parabola on the graph </li></ul><ul><li>A </li></ul><ul><li>B </li></ul><ul><li>C </li></ul>
6. 6. Transformations of f(x) = a(x-h) ² + k <ul><li>State the vertex for the graph of each quadratic and whether it opens up or down: </li></ul><ul><li>1. </li></ul><ul><li>2. </li></ul><ul><li>3. </li></ul>
7. 7. Transformations using calculator <ul><li>Substitute numbers in for all variables a, h, and k.  Put this equation in for y1 and make additional equations for y2, y3 by changing just one of the variables as described. Watch how the graph changes. </li></ul><ul><li>Function f(x) = ax² + bx + c, what happens to the graph of f(x) when c decreases? (This is in standard form, not vertex form) </li></ul>
8. 8. Transformations using function notation f(x) and g(x) <ul><li>EQ: g(x) = f(x) +1 EQ: g(x) = f(x-3) </li></ul><ul><li>What are the transformations? </li></ul><ul><li>EQ: g(x) = 5f(x) </li></ul><ul><li>EQ: g(x) = f(x+8) + 4 </li></ul><ul><li>EQ: g(x) = f(x) – 7 </li></ul>
9. 9. Standard Form & Vertex EX: Find the vertex of f(x) = x² + 8x + 3 and state whether it is a maximum or minimum point.
10. 10. Standard Form & Vertex <ul><li>Use the graph of the function </li></ul><ul><li>y = 3x² - 6x – 1 to state: a. the equation for the axis of symmetry b. the y -intercept (written as an ordered pair) c. the reflection point that corresponds to the y -intercept </li></ul>
11. 11. Applications of Quadratics <ul><li>Question [7.3] A cell phone manufacturer makes profits (P) depending on the sale price (s) of each phone.  The function P= -s² + 120s – 2000 models the monthly profit for a flip phone from Cellular Heaven. What should the phone price be to make the maximum profit?  What is the maximum profit possible? </li></ul><ul><li>Steps to solve this application problem: 1. Find the vertex (x, y)= (x= sale price of phone, y= Profit on phone sales) 2. The x-coordinate is the price of the phone when the profit is a maximum 3. The y-coordinate is the maximum possible profit </li></ul>
12. 12. Applications of Quadratics <ul><li>Question [7.3] For a model rocket, the altitude (h, in meters) as a function of time (t, in seconds) is given by the formula h = 68t – 8t². Use your graphing calculator to find:       a. What is the height of the rocket 2 seconds after it is launched?       b. How long does it take the rocket to reach 100 meters? Round your answer(s) to the nearest hundredth of a second. Steps to solve this problem: 1. For (a), just substitute the value of t, time, into the equation and solve for h, height. 2. For (b), substitute the value of h, height, into the equation. To solve for t, put the equation into the graphing calculator (y1=left side, y2 =right side) and find the intersection, time </li></ul>
13. 13. Factoring <ul><li>To factor a quadratic expression: </li></ul><ul><li>1. Factor using the Greatest Common Factor .        **GCF(     ) </li></ul><ul><li>2. Factor using the Difference of Squares , if possible.        **x²-y²=(x+y)(x-y) </li></ul><ul><li>3. Factor each Trinomial into 2 binomials, if possible.      **ax²+bx+c=(  )(  ) </li></ul><ul><li>4. Factor each quadratic completely.   They may factor more than one time, so try to factor your answer again. </li></ul><ul><li>EX:Factor </li></ul><ul><li>1. 9-18x+81x² </li></ul><ul><li>2. 81n²-49 </li></ul><ul><li>3. 3x²-21x </li></ul><ul><li>4. 7z²-7 </li></ul><ul><li>5. 2x²+5x-12 </li></ul>