Quiz 4-1, What is the vertex of: 2. What is the vertex of:

Intercept Form and Solving by factoring. Intercept Form and Solving by factoring. CP Math Lesson #4-3

Vocabulary Quadratic Function: A function that when graphed is a parabola. Standard Form: Vertex Form: Intercept Form:

Intercept form y = 0 x = -1 x = -1 Graph the following on your calculator: on your calculator: What are the x-intercepts? x = +2 x = +2

Vocabulary Intercept Form: Opens up if positive ‘x-intercepts are: ‘p’ and ‘q’ ‘x-intercepts are: ‘+1’ and ‘+3’ ‘x-intercepts are: ‘-2’ and ‘-4’ Opensdown

Intercept form y = 0 x = -1 x = -1 Why do the intercept have the opposite sign? have the opposite sign? (x + 1) equals some number. x = +2 x = +2 (x – 2) equals another number. These two numbers multiplied together equal 0. (x + 1) = 0 (x – 2) = 0 x = -1 x = -1 x = +2 x = +2

Vocabulary Zero Product Property: If the product of 2 numbers equals 0, A * B = 0 then either: A = 0 and/or B = 0. Then by the zero product property:

Your turn: Which direction does it open and what are the Which direction does it open and what are the x-intercepts of the the following parabolas: x-intercepts of the the following parabolas:

Finding the vertex: If you know the x-intercepts, how do you find the axis of symmetry? Half way between the x-intercepts. x-intercepts are: 4, 6 Axis of symmetry is: x = 5 If you know the axis of symmetry, how do you find the x-coordinate of the vertex? Same as the axis of symmetry x = 5 If you know the x-coordinate of the vertex, how do you find the y-coordinate? The vertex is: (5, 4)

Your turn: Find the vertex of the parabola: 4. 5.

Vocabulary Monomial: an expression with one term. Binomial: expression with two unlike terms. The sum (or difference) of 2 unlike monomials.

Vocabulary Trinomial: expression with three unlike terms. The sum of 3 unlike monomials Or the product of 2 binomials. Intercept form is the product of 2 binomials!!

Standard Form vs. Intercept Form How do we find the axis of symmetry?: How can we find the vertex ? vertex How can we find the x-intercepts? Graph it OR somehow convert it to intecept form.

Vocabulary To Factor: split a binomial, trinomial (or any “nomial”) into its original factors. “nomial”) into its original factors. Standard form: Factored form: Intercept form is a standard form that has been factored.

In order to Factor you must: Know how to multiply two binomials (x – 5)(x + 1) = ?

Smiley Face I call this method the “smiley face”. (x – 4)(x + 2) = ? Left-most term  left “eyebrow” right-most term  right “eyebrow” “nose and chin” combine to form the middle term. You have learned it as FOIL.

Your turn: Multiply the binomials: 6. (x + 2)(x + 1) 7. (x -4)(x + 3)

Factoring Quadratic expressions: (x – 5)(x + 1) (_ + _)(_ + _)

Factoring Quadratic expressions: (x – 5)(x + 1) = ? (x + _)(x + _) -1, 5 5, -1 -5, 1 1, -5 These are the same if lead coefficient is ‘1’ if lead coefficient is ‘1’ These are the same if lead coefficient is ‘1’ if lead coefficient is ‘1’ -1, 5 1, -5

Factoring Quadratic expressions: (x – 5)(x + 1) = ? (x + _)(x + _) -1, 5 1, -5 (x – 1)(x + 5) (x – 5)(x + 1)

(x m)(x n) c = mn (x + 3)(x + 2) Factoring What 2 numbers when multiplied equal 6 and when added together 5? and when added together 5? b = n + m

(x m)(x n) (x – 5)(x + 1) Factoring What 2 numbers: added equal -4 added equal -4 multiplied equal -5?

Your Turn: (x – 5)(x + 1) = ? Factor:

They come in 4 types: (x + 3)(x + 1) Both positive 1 st Negative, 2 nd Positive (x – 1)(x – 5) 1 st Positive, 2 nd Negative (x + 8)(x – 2) Both negative (x – 4)(x + 2)

Your Turn: (x – 5)(x + 1) = ? Factor:

Special Products Product of a sum and a difference. (x + 2)(x – 2) “conjugate pairs” (x + 2)(x – 2) “nose and chin” are additive inverses are additive inverses of each other. of each other. “The difference of 2 squares.”

Your turn: Multiply the following conjugate pairs: 13. (x – 3)(x + 3) 14. (x – 4)(x + 4) “The difference of 2 squares.” “The difference of 2 squares” factors as conjugate pairs.

Your Turn: Factor:

Special Products Square of a sum. (x + 2)(x + 2)

Special Products Square of a sum. (x + 3)(x + 3)

Special Products Square of a difference. (x - 4)(x - 4)

Special Products Square of a difference. (x - 3)(x - 3)

Your Turn: Simplify (multiply out) We now have all the tools to “solve by factoring”

Vocabulary Quadratic Equation: Root of an equation: the x-value where the graph crosses the x-axis (y = 0). crosses the x-axis (y = 0). Zero of a function: same as root Solution of a function: same as both root and zero of the function. x-intercept: same as all 3 above.

Solve by factoring (1) factor the quadratic equation. (1) factor the quadratic equation. (2) set y = 0 (3) Use “zero product property” to find the x-intercepts

Your Turn: Solve by factoring:

Your turn: Multiply the binomials: Multiply the binomials: 23. (2x – 1)(x + 3) 24. (x + 5)(x – 5) Factor the quadratic expressions: