4. (a) Write the equation of the parabola with vertex (4,3) and Directrix x = 2.

Transcription

1 Station 1 Completing the Square, Vertex Form, the Directrix and Focus 1. Find the center and radius of the circle x + 14x + y 9y + 5 = 0. (a) Graph the equation y = x + 7x + 1 (b)write the equation in vertex form (c) What is the vertex? (d) What are the x intercepts? 3. For what value of k will x + 5x k=0 have exactly one solution? 4. (a) Write the equation of the parabola with vertex (4,3) and Directrix x =. (b) What is the vertex, directrix, and focus of the parabola with equation 1y = (x 3)?

2 Station 1 Answers 1. Find the center and radius of the circle x + 14x + y 9y + 5 = 0 (x + 7) + (y 9/) = /4 = 69/4 Center (-7, 9/), radius 69. (a) Graph the equation y = x + 7x + 1 (b)write the equation in vertex form y 1 = x + 7x y = x + 7x + 49/4 y + ¼ = (x + 7/) (c) What is the vertex? (-7/, -1/4) (d) What are the x intercepts? y = (x + 3)(x + 4) (-3,0), (-4,0) 3. For what value of k will x + 5x k=0 have exactly one solution? 5 + 8k = 0 k = -5/8 4. (a) Write the equation of the parabola with vertex (4,0) and Directrix x =. p = 8(x 4) = (y 3) (b) What is the vertex, directrix, and focus of the parabola with equation 1y = (x 3)? V = (3, 0), faces up with p = 3, F = (3, 3), directrix: y = -3

3 Station The quadratic formula and the discriminant 1. Find the constant k for which x + x 5 + k has exactly one zero.. If x = is a zero of x 3 + 6x 19x + 6, factor the polynomial over the reals. 3. Find the domain of the function y x x = Find the x-intercepts of y = x - 7x Use the x-intercepts to find the vertex of the parabola.

4 Station Answers 1. Find the constant k for which x + x 5 + k has exactly one zero. Complete the square: 5 = 5 4. If x = is a zero of x 3 + 6x 19x + 6, factor the polynomial over the reals. If x = is a zero, (x ) is a factor. Use long division to find x + 8x 3 as the other factor. Using the quadratic formula, we get x = and x = -4-19, so the factors are (x )(x ( ))(x (-4-19 )). 3. Find the domain of the function y = x + x 1 x + x 1 > 0 ± 8 = 1± are the zeroes of x + x 1. By a sign graph we can see that: (, 1 ] [ 1+, ) 4. Find the x-intercepts of y = x - 7x Use the x-intercepts to find the vertex of the parabola. (x )(x 5) = 0 (, 0) and (5,0). The midpoint between these points is (3.5,0) (Note that this is b/a). Therefore the vertex is (3.5, f(3.5)) or (3.5, -.5)

5 Station 3 Graphs 1. (a) Write y = -x + x + 8 in vertex form (b) Graph the parabola. (c) What is the axis of symmetry? (d) What are the x-intercepts?. Find the equation of a parabola in y = ax + bx + c form of a parabola that has axis of symmetry x = - ¼, y-intercept 4, and contains (-1, 3) 3. (without a calculator) Sketch a Graph of y = x 5 4. For what values of x is the graph of y = -x + 4x above the x-axis?

6 Station 3 Answers 1. (a) Write y = -x + x + 8 in vertex form y 8 = - (x x) y 8-1 = - (x x + 1) y 9 = - (x 1) (b) Graph the parabola. (c) What is the axis of symmetry? x = 1 (d) What are the x-intercepts? (,0) or (-4,0). Find the equation of a parabola in y = ax + bx + c form of a parabola that has axis of symmetry x = - ¼, y-intercept 4, and contains (-1, 3) c = 4, -b/a = - ¼, and 3 = a b + c = a b + 4 a b = -1 4b = a or b = a b b = -1, so b = -1 and a = y = -x x (without a calculator) Sketch a Graph of y = x 5 4. For what values of x is the graph of y = -x + 4x above the x-axis? - x + 4x > 0 x 4x + < 0 4 ± 8 x = = ±. By a sign graph we see the solution interval is (, + )

7 Station 4 Word Problems; sums and products of roots 1. A different scenario: you decide to start a neighborhood garden and you have to fence it on all four sides. (It s still a rectangle.) Mayor Daley s fascination with wrought-iron fences means that one side will cost $0 per foot, while the other three sides will cost $15 per foot. If you have $100 with which to fence the garden, what are the dimensions of the largest possible garden?. Find the length of side s of a square if the area of the square is numerically the same as three more than s. 3. If r and s are the zeroes of x + 5x 8, compute a. r + s b. rs c. r + s d r s 4. What is the sum of the roots of the equation x x x = 0?

8 Station 4 Answers 1. A different scenario: you decide to start a neighborhood garden and you have to fence it on all four sides. (It s still a rectangle.) Mayor Daley s fascination with wrought-iron fences means that one side will cost $0 per foot, while the other three sides will cost $15 per foot. If you have $100 with which to fence the garden, what are the dimensions of the largest possible garden? Let the expensive side be x feet long and the other side be y feet long. Then 0x + 15x + 15y + 15y = x y = 100 y = 40 7/5 x. The area of the rectangle is xy = x(40 7/5x), and the maximum occurs at x = 14/5 = 50 or about 7.14 feet; y = 30 feet. 7. Find the length of side s of a square if the area of the square is numerically the same as three more than s. s = 3 + s s s 3 = s = 3. If r and s are the zeroes of x + 5x 8, compute a. r + s b. rs c. r + s d r s a. r + s = -5/ b. rs = -4 c. r + s = (r + s) rs = (-5/) (-4) = 5/4 + 8 = 57/4 d. 1 1 r+ s + = = 5/ = 5 r s rs What is the sum of the roots of the equation x x x = 0? In any nth degree polynomial, the sum of the roots is the coefficient of the n -1th degree term divided by the leading coefficient. In this case, n 1 = 4. There is no 4 nd degree term so the sum of the roots is zero.