Quadratic Functions Chapter 7

Vertex Form Vertex (h, k)

Vertex Form a > 0, opens upward a < 0, opens downward the larger│a│is the narrower the parabola the closer a is to zero the wider the parabola

Stretching the Unit Quadratic

Reflecting Across the x-axis

Translating Graphs Up/Down

Translating Graphs Right/Left

Graphing a Quadratic Function First graph vertex Find a point

Draw axis of symmetry through vertex Reflect point over axis Graphing a Quadratic Function

Finding a Quadratic Model Create a scattergram Select a vertex (Doesn’t have to be data point) Select non-vertex point Plug vertex in for h and k, and the nonvertex point for x and f(x)/y into a standard equation Solve for a Then substitute a into the standard equation

Graph Quadratic Model Pick vertex –(70, 5) Pick point –(40, 9) xf(x) 1930 (30) (40) (50) (60) (70) (80) (90) (100)10

7.2 Graphing Quadratics in Standard Form

Quadratic in Standard Form Find y-intercept (0, c) Find symmetric point Use midpoint formula of the x-coordinates of the symmetric points to find the x- coordinate of the vertex Plug x-coordinate of the vertex into equation for x

Graphing Quadratics Y-intercept –(0, 7) Symmetry Point

Graphing Quadratics (0, 7) (6, 7) Midpoint

Vertex formula vertex formula x-coordinate y-coordinate

Vertex Formula

Maximum/Minimum For a quadratic function with vertex (h, k) If a > 0, then the parabola opens upward and the vertex is the minimum point (k minimum value) If a < 0, then the parabola opens downward and vertex is the maximum point (k maximum value)

Maximum Value Model A person plans to use 200 feet of fencing and a side of her house to enclose a rectangular garden. What dimensions of the rectangle would give the maximum area? What is the area?

Maximum area would be 50 x 100 = 5000

7.3 Square Root Property

Product/Quotient Property for Square Roots For a ≥ 0 and b ≥ 0, For a ≥ 0 and b > 0, Write radicand as product of largest perfect-square and another number Apply the product/quotient property for square roots

Simplifying Radical Expressions No radicand can be a fraction No radicand can have perfect-square factors other than one No denominator can have a radical expression

Examples

Square Root Property Let k be a nonnegative constant. Then, is equivalent to

Imaginary Numbers Imaginary unit, (i), is the number whose square is -1. Square root of negative number –If n is a positive real number,

Complex Numbers A complex number is a number in the form Examples Imaginary number is a complex number, where a and b are real numbers and b ≠ 0

Solving with Negative Square Roots

7.4 Completing the Square

Perfect Square Trinomial For perfect square trinomial in the form dividing by b by 2 and squaring the result gives c:

Examples

7.5 Quadratic Formula

Quadratic Formula The solutions of a quadratic equation in the formare given by the quadratic formula:

Determining the Number of Real- Number Solutions The discriminant isand can be used to determine the number of real solutions If the discriminant > 0, there are two real- number solutions If the discriminant = 0, there in one real- number solution If the discriminant < 0, there are two imaginary-number solutions (no real)

Quadratic Formula

Examples One real-number solutionTwo imaginary-number solutions

Intersections with y = n lines/points at a certain height Note if the discriminant is < 0, then there are no intersections