Unit 4 Lesson 2:Solving Quadratic Equations by Graphing Advanced Math Topics Mrs. Mongold

Parts of a Quadratic Equation y = ax2 + bx + c

Parts of a Quadratic Equation y = ax2 + bx + c ax2 is the quadratic term.

Parts of a Quadratic Equation y = ax2 + bx + c ax2 is the quadratic term. bx is the linear term.

Parts of a Quadratic Equation y = ax2 + bx + c ax2 is the quadratic term. bx is the linear term. c is the constant term.

Parts of a Quadratic Equation y = ax2 + bx + c ax2 is the quadratic term. bx is the linear term. c is the constant term. The highest exponent is two; therefore, the degree is two and there are two solutions

Identifying Terms Example f(x)=5x2-7x+1

Identifying Terms Example f(x)=5x2-7x+1 Quadratic term 5x2

Identifying Terms Example f(x)=5x2-7x+1 Quadratic term 5x2 Linear term -7x

Identifying Terms Example f(x)=5x2-7x+1 Quadratic term 5x2 Linear term -7x Constant term 1

Identifying Terms Example f(x) = 4x2 - 3

Identifying Terms Example f(x) = 4x2 - 3 Quadratic term 4x2

Identifying Terms Example f(x) = 4x2 - 3 Quadratic term 4x2 Linear term 0

Identifying Terms Example f(x) = 4x2 - 3 Quadratic term 4x2 Linear term 0 Constant term -3

Quadratic Solutions The number of real solutions is at most two. No solutions One solution Two solutions

Quadratic Solutions The number of real solutions is at most two. No solutions One solution Two solutions Number of real solutions is indicated by how many times the graph intercepts x axis

Solving Equations When we talk about solving these equations, we want to find the value of x when y = 0. These values, where the graph crosses the x-axis, are called the x-intercepts. These values are also referred to as solutions, zeros, or roots.

Identifying Solutions Example f(x) = x2 - 4

Identifying Solutions Example f(x) = x2 – 4 replace y or f(x) with 0 and solve 0 = x2 - 4

Identifying Solutions Example f(x) = x2 – 4 replace y or f(x) with 0 and solve 0 = x2 – 4 Square root property says solutions are x = 2 and x = - 2

Identifying Solutions Example f(x) = x2 – 4 replace y or f(x) with 0 and solve 0 = x2 – 4 Square root property says solutions are x = 2 and x = - 2 Two points aren’t enough we need 5

To get 3 more we could make a table or x/y chart….but that’s not fun!

So we are going to get our equation in standard form (which is fun) and faster/easier!

Standard Form y= a(x – h)2 + k Vertex (h, k) Axis of Symmetry x = h a tells us how fat/skinny and if parabola is opening up or down

Identifying Solutions Example f(x) = x2 – 4 Standard Form: y = (x - 0)2 – 4 Vertex (0, -4) Solutions are -2 and 2 so we graph (-2, 0) and (2, 0)

Now we have 3 points we only need 2 more Use x/y chart one time and the AOS will get us the 5th point x = 1 gives us y = -3

Now graph Vertex (0, -4) Solutions (-2,0) and (2,0) Point (1, -3) Use AOS x = 0 to get 5th point on parabola (-1, -3)

Identifying Solutions Now you try this problem. f(x) = 2x - x2

Identifying Solutions Now you try this problem. f(x) = 2x - x2 Solutions are 0 and 2.

Graphing Quadratic Equations The graph of a quadratic equation is a parabola. The roots or zeros are the x-intercepts. The vertex is the maximum or minimum point. All parabolas have an axis of symmetry.

Graphing Quadratic Equations One method of graphing uses a table with arbitrary x-values. Graph y = x2 - 4x

Graphing Quadratic Equations One method of graphing uses a table with arbitrary x-values. Graph y = x2 - 4x Roots 0 and 4 , Vertex (2, -4) , Axis of Symmetry x = 2

Graphing Quadratic Equations Try this problem y = x2 - 2x - 8. Roots Vertex Axis of Symmetry

Graphing Quadratic Equations The graphing calculator is also a helpful tool for graphing quadratic equations. We will talk about this tomorrow HW worksheet #1