Solving Quadratic Equations by Graphing Section 9 – 2

Main Ideas Solve quadratic equations by graphing. Estimate solutions of quadratic equations by graphing.

Quadratic Equation: an equation of the form ax2 + bx + c = 0 Vocabulary Quadratic Equation: an equation of the form ax2 + bx + c = 0 ,where a ≠ 0.

Vocabulary The solutions of a quadratic equation are called the____________________. The roots of a quadratic equation can be found by graphing the related quadratic function f(x) = ax2 + bx + c and finding the _____________________________.

d.) Graph y = x2 + 4x + 3 X Y -4 3 -3 -2 -1 Vertex 

f.) Solve x2 + 4x + 3 = 0 by graphing To solve you need to know where the value of f(x) = 0. This occurs at the x-intercepts, (-3,0) and (-1,0). The solutions are x = {-3, -1}

d.) Graph x2 + 7x + 12 = 0 X Y -5 2 -4 -3.5 -0.25 -3 -2 Vertex 

f.) Solve x2 + 7x + 12 = 0 by graphing To solve you need to know where the value of f(x) = 0. This occurs at the x-intercepts, (-3,0) and (-4,0). The solutions are x = {-4, -3}

d.) Graph x2 – 4x + 5 = 0 X Y 5 1 2 3 4 Vertex 

f.) Solve x2 – 4x + 5 = 0 by graphing To solve you need to know where the value of f(x) = 0. There is no x- intercepts, so there is no real roots Solution: {Ø}

d.) Graph x2 – 6x + 9 = 0 X Y 1 4 2 3 5 Vertex 

f.) Solve x2 – 6x + 9 = 0 by graphing To solve you need to know where the value of f(x) = 0. This occurs at the x-intercepts (3,0). Solution: {3}

Solving Quadratic Equations by Graphing Section 9 – 2 Day 2

Main Ideas Solve quadratic equations by graphing. Estimate solutions of quadratic equations by graphing.

Estimate Solutions The roots of a quadratic equation may not be integers. If exact roots cannot be found, they can be estimated by finding the consecutive integers between which the roots lie.

Example 2: Solve the equation by graphing x2 + 6x + 6 = 0 a.) Write the equation of the axis of symmetry b.) Find the coordinates of the vertex of the graph of each function. c.) Identify the vertex as a maximum or a minimum. d.) Then graph the function. e.) State the Domain and Range f.) Determine what the x-intercept or zeros are to solve the quadratic equation.

Solve x2 + 6x + 6 = 0 by graphing a.) equation of axis of symmetry a = _____ b = ____ c = ____ b.) Find the vertex. 1 6 6 y = x2 + 6x + 6 when x = -3 y = (-3)2 + 6(-3) + 6 y = 9 + -18 + 6 y = -3 Vertex (-3, -3)

Solve x2 + 6x + 6 = 0 by graphing c.) Identify the vertex as a maximum or a minimum. Since the coefficient of the x2-term is positive, the parabola opens upward, and the vertex is a minimum point.

d.) Graph x2 + 6x + 6 = 0 X Y -5 1 -4 -2 -3 -1 Vertex 

Domain: D: {x I x is all real numbers.} Range: R: {y I y ≥ -3} e.) Graph x2 + 6x + 6 = 0 Domain: D: {x I x is all real numbers.} Range: R: {y I y ≥ -3}

f.) Solve x2 + 6x + 6 = 0 by graphing The x-intercepts of the graph are between -5 and -4 and between -2 and -1. So one root is between 5 and 4, and the other root is between 2 and 1. -5 < x < -4 -2 < x < -1

Try 4: Solve the equation by graphing x2 – 2x – 4 = 0 a.) Write the equation of the axis of symmetry b.) Find the coordinates of the vertex of the graph of each function. c.) Identify the vertex as a maximum or a minimum. d.) Then graph the function. e.) Determine what the x-intercept or zeros are to solve the quadratic equation.

Solve x2 – 2x – 4 = 0 by graphing a.) equation of axis of symmetry a = ____ b = ____ c = ____ b.) Find the vertex. 1 -2 -4 y = x2 – 2x – 4 when x = 1 y = (1)2 + -2(1) + -4 y = 1 + -2 + -4 y = -5 Vertex (1, -5)

Solve x2 – 2x – 4 = 0 by graphing c.) Identify the vertex as a maximum or a minimum. Since the coefficient of the x2-term is positive, the parabola opens upward, and the vertex is a minimum point.

d.) Graph x2 – 2x – 4 = 0 X Y -1 -4 1 -5 2 3 Vertex 

e.) Domain and Range Domain: D: {x I x is all real numbers.} Range: R: {y I y ≥ -5}

Solve x2 + 6x + 6 = 0 by graphing The x-intercepts of the graph are between -2 and -1 and between 3 and 4. So one root is between - 2 and -1, and the other root is between 3 and 4. -2 < x < -1 3 < x < 4

2 roots x = { -4, 3} x – 3 = 0 x + 4 = 0 x + -3 + 3 = 0 + 3 Use factoring to determine how many times the graph of each function intersects the x-axis. Identify the root. f(x) = x2 + x – 12 0 = x2 + x – 12 0 = (x – 3)(x + 4) x – 3 = 0 x + -3 + 3 = 0 + 3 x = 3 x + 4 = 0 x + 4 + -4 = 0 + -4 x = - 4 2 roots x = { -4, 3}