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1. 1. Solving Quadratic Equations An explanation by Molly Murphy
2. 2. Forms of Quadratics <ul><li>Standard Form </li></ul><ul><ul><ul><ul><li>ƒ(x) = ax² + bx + c </li></ul></ul></ul></ul><ul><li>Vertex Form </li></ul><ul><ul><ul><ul><li>ƒ(x) = a(x - h)² + k </li></ul></ul></ul></ul><ul><li>Intercept Form </li></ul><ul><ul><ul><ul><li>ƒ(x) = a(x - p)(x - q) </li></ul></ul></ul></ul>
3. 3. Role of “a” in quadratics <ul><li>If a>0 the parabola opens up </li></ul><ul><li>If a<0 the parabola opens down </li></ul><ul><li>Graph is narrow if |a|>1 </li></ul><ul><li>Graph is wide if |a|<1 </li></ul>
4. 4. Standard Form (finding vertex) <ul><li>The vertex is (-b/2a , ƒ(-b/2a)) </li></ul><ul><li>The axis of symmetry is the vertical line going through the vertex. It is written algebraically as x=-b/2a (because –b/2a is the x-coordinate of the vertex) </li></ul>
5. 5. Standard Form (max. and min.) <ul><li>The vertex is the parabola’s maximum or minimum value. </li></ul><ul><li>If the parabola opens up it has a minimum. If it opens down it has a maximum. </li></ul>
6. 6. Standard Form (role of c) <ul><li>“c” is the y-intercept of the parabola. The point (0 , c) is on the parabola. </li></ul>
7. 7. Graphing in Standard form <ul><li>Example: ƒ(x) = 2x² + 4x + 1 </li></ul><ul><li>Find the x-coordinate of the vertex </li></ul><ul><li>-b/2a = -4/2(2) = -4/4= -1 </li></ul><ul><li>Plug that number back into the function to find the y-coordinate </li></ul><ul><li>2(-1)²+4(-1)+1 = -2²-4+1 = 2-4+1 = -1 </li></ul><ul><li>The vertex is (-1 , -1) </li></ul><ul><li>Make a chart with other values on either “side” of the x-coordinate of the vertex. Plug the x values into the function to get the y values. </li></ul>7 1 -1 1 7 Y 1 0 -1 -2 -3 X
8. 8. Continued <ul><li>Plot the points on the graph and draw parabola. </li></ul><ul><li>Check with graphing calculator. </li></ul>(-3 , 7) (1, 7) (0 , 1) (-2 , 1) (-1 , -1)
9. 9. Vertex Form (finding vertex) <ul><li>ƒ(x) = a(x - h)² + k </li></ul><ul><li>The vertex is (h , k) </li></ul>The vertex is (0 , 1) So h=0 and k=1
10. 10. Graphing in Vertex Form <ul><li>ƒ(x) = 2(x + 1)² - 1 </li></ul><ul><li>We already know the vertex is (-1 , -1) </li></ul><ul><li>Make a chart of x and y values. Plug the x values into the equation and solve to get the y values. </li></ul>7 1 -1 1 7 Y 1 0 -1 -2 -3 X
11. 11. Continued <ul><li>Plot the vertex and other points on the graph. Draw parabola. </li></ul><ul><li>Check with graphing calculator. </li></ul>(-3 , 7) (1 , 7) (-2 , 1) (0 , 1) (-1 , -1)
12. 12. Intercept Form (finding vertex) <ul><li>ƒ(x) = a(x – p)(x – q) </li></ul><ul><li>For the x-coordinate of the vertex, average p and q (p+q)/2. For the y-coordinate of the vertex, plug that value back into the function. ƒ((p+q)/2). </li></ul>
13. 13. Graphing in Intercept Form <ul><li>ƒ(x) = a(x – p)(x – q) </li></ul><ul><li>The x-intercepts are p and q. </li></ul><ul><li>ƒ(x) = 2(x + 3)(x + 1) </li></ul><ul><li>Plot the vertex (-3+-1)/2 = -4/2 = -2 </li></ul><ul><li>y = 2(-2+3)(-2+1) = -2 the vertex is (-2 , -2) </li></ul><ul><li>Next plot the x-intercepts (0 , -3) and (0 , -1) </li></ul><ul><li>Draw in parabola. </li></ul>
14. 14. Continued <ul><li>Check with a graphing calculator. </li></ul>(0 , -1) (0 , -3) (-2 , -2)