Factor and Solve Quadratic Equations Ms. Nong

What is in this unit? Graphing the Quadratic Equation Identify the vertex and intercept(s) for a parabola Solve by taking SquareRoot & Squaring Solve by using the Quadratic Formula Solve by Completing the Square Factor & Solve Trinomials (split the middle) Factor & Solve DOTS: difference of two square Factor GCF (greatest common factors) Factor by Grouping

Parts of a Parabola The ROOTS (or solutions) of a polynomial are its x-intercepts Recall: The x-intercepts occur where y = 0. Roots ~ X-Intercepts ~ Zeros means the same

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. One solution X = 3 Two solutions X= -2 or X = 2 No solutions

Vertex (h,k) Maximum point if the parabola is up-side-down Minimum point is when the Parabola is UP a>0 a<0

All parts labeled

Can you answer these questions? How many Roots? Where is the Vertex? (Maximum or minimum) What is the Y-Intercepts?

What is in this unit? Graph the quadratic equations (QE) Solve by taking SquareRoot & Squaring Solve by using the Quadratic Formula Solve by Completing the Square Factor & Solve Trinomials (split the middle) Factor & Solve DOTS: difference of two square Factor GCF (greatest common factors) Factor by Grouping

Finding the Axis of Symmetry When a quadratic function is in standard form y = ax2 + bx + c, the equation of the Axis of symmetry is This is best read as … ‘the opposite of b divided by the quantity of 2 times a.’ 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? The Axis of symmetry is x = 3. a = 3 b = -18

The x-coordinate of the vertex is 2 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 The x-coordinate of the vertex is 2 a = -2 b = 8

The vertex is (2 , 5) Finding the Vertex 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. The vertex is (2 , 5)

Graphing a Quadratic Function STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. y x 3 2 y x –1 5

Y-intercept of a Quadratic Function Y-axis y x The y-intercept of a Quadratic function can Be found when x = 0. The constant term is always the y- intercept

Example: Graph y= -.5(x+3)2+4 a is negative (a = -.5), so parabola opens down. Vertex is (h,k) or (-3,4) Axis of symmetry is the vertical line x = -3 Table of values x y -1 2 -2 3.5 -3 4 -4 3.5 -5 2 Vertex (-3,4) (-4,3.5) (-2,3.5) (-5,2) (-1,2) x=-3

Your assignment: