Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Cubic function wikipedia , lookup
Factorization wikipedia , lookup
History of algebra wikipedia , lookup
Signal-flow graph wikipedia , lookup
Elementary algebra wikipedia , lookup
Quartic function wikipedia , lookup
Median graph wikipedia , lookup
Vertex operator algebra wikipedia , lookup
Graph the following transformations f ( x) ( x 2) 3 f ( x) 2( x 1) 4 2 Warm - Up 2 Assignment ◦ p. 214 ◦ #12, 13, 16, 17, 20, 22, 24, Cargo Pants Legends Make Up Tests ◦ After School Today and Tomorrow Test Corrections ◦ After School Wednesday and Thursday Announcements 2.1 Quadratic Functions Section Objectives: Students will know how to sketch and analyze graphs of quadratic functions. Quadratic functions f(x) = ax2 + bx + c, with a 0. The graph of a quadratic function is a parabola. y = x2 ◦ Vertex? y = -x2 ◦ Vertex? The Graph of a Quadratic Function The quadratic function standard form f(x) = a(x – h)2 + k, a 0 Vertex at (h, k). If a > 0, the parabola opens _______ If a < 0, the parabola opens _______ The Standard Form of a Quadratic Function Is it a perfect square?? Yes ◦ Factor/Use Quadratic Formula to find the vertex. x 2x 1 2 •No •Complete the Square to find the vertex x 3x 4 Changing a trinomial into vertex form 2 ax bx c 2 The vertex is at 2 (2, -6), the parabola f ( x) x 4 x 2 opens upward, and 2 f ( x) x 4 x 2 there is no change in 2 x 4 x 4 4 2 the width. x 2 6 2 Warm Up - Graph the following quadratic function. Assignment ◦ p. 214 ◦ # 32, 34, 37, 38, 39, 44, 70 Make Up Tests ◦ After School Today Test Corrections ◦ After School Tomorrow and Thursday Announcements p. 214 #12, 13, 16, 17, 20, 22, 24, Assignment Questions? f ( x) x 8x 16 2 f ( x) x 2 8 x 16 x 4x 4 x 4 2 Vertex: (4, 0) Find the vertex of the following parabola. •To find the x – intercept: •To find the y – intercept: •Set “y” equal to 0 •Set “x” equal to 0 •Solve for x •Solve for y y x 5 6 2 y x 5 6 2 0 x 5 6 2 6 x 5 5 6 x y 0 5 6 2 y 5 6 2 2 6 x5 y x 5 6 2 y 25 6 (5 6, 0) y 19 (0,19) Find the x and y intercepts y = (x – 4)2 + 5. ◦ What is the minimum value of y? Finding Minimum and Maximum Values of Quadratics From the vertex we have this much of the equation: f(x) = a(x – 1)2 – 2. To find a we substitute the point (3, 6) and solve for a. 2 6 a3 1 2 6 4a 2 8 4a 2a The equation is f(x) = 2(x – 1)2 – 2. Example 3. Find the standard form of the equation of the parabola that has vertex (1, -2) and passes through the point (3, 6). Example 4. The daily cost of manufacturing a particular product is given by C(x) = 1200 – 7x + 0.1x2 where x is the number of units produced each day. Determine how many units should be produced daily to minimize cost. Algebraic Solution Graphical Solution We need to find h. C ( x) 1200 7 x 0.1x 2 0.1 x 2 70 x 1200 0.1 x 2 70 x 352 1200 122.5 2 0.1x 35 1322.5 Producing 35 units per day will minimize cost. Mo’ Money, Long Problems