Quadratic Equations and Functions Chapter 9 Quadratic Equations and Functions

9.1 Graph y = ax2 + c (b = 0)

Quadratic Functions parabola The graph of a quadratic function is a: y x A parabola can open up or down. Vertex If the parabola opens up, the lowest point is called the vertex (minimum). Let students know that in Algebra I we concentrate only on parabolas that are functions; In Algebra II, they will study parabolas that open left or right. If the parabola opens down, the vertex is the highest point (maximum). Vertex NOTE: if the parabola opens left or right it is not a function!

y = ax2 + bx + c The standard form of a quadratic function is: The parabola will open up when the a value is positive. a < 0 a > 0 The parabola will open down when the a value is negative. Remind students that if ‘a’ = 0 you would not have a quadratic function.

The Axis of symmetry ALWAYS passes through the vertex. Parabolas are symmetric. If we drew a line down the middle of the parabola, we could fold the parabola in half. y x Axis of Symmetry We call this line the Axis of symmetry. The Axis of symmetry ALWAYS passes through the vertex.

Steps to Graphing Quadratic Functions 1) Find the Axis of symmetry using: 2) Find the vertex by using x to find y 3) Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve.

Graph y = x2 (parent function)

Graph y = -3x2 y x

Graph y = ½x2 + 1 y x

Homework: 9.1 Practice

9.2 Graph y = ax2 + bx + c

Find the Axis of symmetry for y = 3x2 – 18x + 7 y = ax2 + bx + c, When a quadratic function is in standard form the equation of the Axis of symmetry is Find the Axis of symmetry for y = 3x2 – 18x + 7 Discuss with the students that the line of symmetry of a quadratic function (parabola that opens up or down) is always a vertical line, therefore has the equation x =#. Ask “Does this parabola open up or down? a = 3 b = -18 The Axis of symmetry is x = 3.

Vertex X-coordinate Finding the Vertex The Axis of symmetry always goes through the _______. Thus, the Axis of symmetry gives us the ____________ of the vertex. Vertex X-coordinate Find the vertex of y = -2x2 + 8x - 3 STEP 1: Find the Axis of symmetry STEP 2: Substitute the x – value into the original equation to find the y –coordinate of the vertex.

Graphing a Quadratic Function There are 3 steps to graphing a parabola in standard form. STEP 1: Find the Axis of symmetry using: STEP 2: Find the vertex STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. MAKE A TABLE using x – values close to the Axis of symmetry.

Graphing a Quadratic Function y x STEP 1: Find the Axis of symmetry STEP 2: Find the vertex STEP 3: Make table of values around vertex

Example: Graph y = -x2 – 2x + 1

Homework: p.580 #1, 2, 3 – 35odd

9.3 Solve Quadratic Equations by Graphing

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 known as solutions, zeros, or roots.

Solving a Quadratic The number of real solutions is at most two. The x-intercepts (when y = 0) of a quadratic function are the solutions to the related quadratic equation. The number of real solutions is at most two. Remind students that x-intercepts are found by setting y = 0 therefore the related equation would be ax2+bx+c=0. Also state that since the highest degree of a quadratic is 2, then there are at most 2 solutions. For the first graph ask “why are there no solutions?”-- there are no solutions because the parabola does not intercept the x-axis. 2nd and 3rd graph ask students to state the solutions. Additional Vocab may be itroduced: The x-intercepts are solutions, zero’s or roots of the equation. Two solutions X= -2 or X = 2 One solution X = 3 No solutions

Identifying Solutions Find the solutions of 2x - x2 = 0 The solutions of this quadratic equation can be found by looking at the graph of f(x) = 2x – x2 The x-intercepts(or Zeros) of f(x)= 2x – x2 are the solutions to 2x - x2 = 0 Point out to students that the function can also be written as y = -x2+2x. X = 0 or X = 2

Example: Solve the equation x2 + 5x + 6 = 0 by graphing

Homework: p. 589 #1, 2, 3 – 39 odd

9.4 Use Square Roots to Solve Quadratic Equations I can solve a quadratic equation by finding square roots I can solve a problem about a falling object CC9-12.A.REI.4b

When b = 0, the equation becomes y = ax2 + c You can use Square Roots Method to solve the equation (extracting roots)

Steps to Solving Quadratics Using Square Roots Method Make sure b = 0 Get variable term on one side, constant on the other Get variable by itself Take square root of both sides

Example: Solve 2x2 = 8

Example: x2 + 12 = 25 Example: x2 + 12 = 7

Example: A pinecone drops from a tree 150 feet up Example: A pinecone drops from a tree 150 feet up. How long will it take the pinecone to reach the ground.

Homework: p. p.597 #1, 2, 3 – 49 odd, 56,

9.6 Solve Quadratic Equations by Using Quadratic Formula I can solve quadratic equations using the quadratic formula. CC.9-12.A.REI.4b

Quadratic Formula:

Example: Solve 3x2 + 5x = 8

Example: Solve 2x2 – 7 = x

Example: x2 – 8x + 16 = 0

Methods for Solving Quadratic Equations Factoring – good when quadratic eqn can be factored easily Graphing – use when approximate solutions are fine Square roots method – use when b = 0 (no bx term) Quadratic formula – can be used at any time!!! Some methods are better than others depending on situation

Example: tell what method you would use to solve the quadratic equation. Explain. x2 + 5x + 6 = 0

Homework: p.616 #1, 2, 3 – 43 odd, 47, 48

9.6 extended – The Discriminant

9.5 Solve Quadratic Equations by Completing the Square

9.7 Solve Systems of Quadratic Equations

9.8 Compare Linear, Exponential, and Quadratic Functions

9.9 Model Relationships