Solving Quadratic Equations An explanation by Molly Murphy

Forms of Quadratics:

Forms of Quadratics Standard Form ƒ(x) = ax² + bx + c Vertex Form ƒ(x) = a(x - h)² + k Intercept Form ƒ(x) = a(x - p)(x - q)

Role of “a” in quadratics:

Role of “a” in quadratics If a>0 the parabola opens up If a<0 the parabola opens down Graph is narrow if |a|>1 Graph is wide if |a|<1

Standard Form (finding vertex):

Standard Form (finding vertex) The vertex is (-b/2a , ƒ(-b/2a)) 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)

Standard Form (max. and min.):

Standard Form (max. and min.) The vertex is the parabola’s maximum or minimum value. If the parabola opens up it has a minimum. If it opens down it has a maximum.

Standard Form (role of c):

Standard Form (role of c) “c” is the y-intercept of the parabola. The point (0 , c) is on the parabola.

Graphing in Standard form:

Graphing in Standard form Example: ƒ(x) = 2x² + 4x + 1 Find the x-coordinate of the vertex -b/2a = -4/2(2) = -4/4= -1 Plug that number back into the function to find the y-coordinate 2(-1)²+4(-1)+1 = -2²-4+1 = 2-4+1 = -1 The vertex is (-1 , -1) 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. X -3 -2 -1 0 1 Y 7 1 -1 1 7

Continued:

Continued Plot the points on the graph and draw parabola. Check with graphing calculator. (-3 , 7) (1, 7) (0 , 1) (-2 , 1) (-1 , -1)

Vertex Form (finding vertex):

Vertex Form (finding vertex) ƒ(x) = a(x - h)² + k The vertex is (h , k) The vertex is (0 , 1) So h=0 and k=1

Graphing in Vertex Form:

Graphing in Vertex Form ƒ(x) = 2(x + 1)² - 1 We already know the vertex is (-1 , -1) Make a chart of x and y values. Plug the x values into the equation and solve to get the y values. X -3 -2 -1 0 1 Y 7 1 -1 1 7

Continued:

Continued Plot the vertex and other points on the graph. Draw parabola. Check with graphing calculator. (-3 , 7) (1 , 7) (-2 , 1) (0 , 1) (-1 , -1)

Intercept Form (finding vertex):

Intercept Form (finding vertex) ƒ(x) = a(x – p)(x – q) 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).

Graphing in Intercept Form:

Graphing in Intercept Form ƒ(x) = a(x – p)(x – q) The x-intercepts are p and q. ƒ(x) = 2(x + 3)(x + 1) Plot the vertex (-3+-1)/2 = -4/2 = -2 y = 2(-2+3)(-2+1) = -2 the vertex is (-2 , -2) Next plot the x-intercepts (0 , -3) and (0 , -1) Draw in parabola.

Continued:

Continued Check with a graphing calculator. (0 , -1) (0 , -3) (-2 , -2)

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