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Quadratic Brochure

Vocabulary

Vertex - a meeting point of two lines that form an angle.

Quadratic - a quadratic equation.

Intersection - the point where two or more lines meet.

Parabola - a symmetrical open plane curve.

Vertex form - f(x)=a(x-h)^2+k

Standard form - most commonly written form.

Axis of symmetry - line of symmetry for the graph. All points on either side of axis are symmetrical.

Vocabulary Continued

Quadratic formula - the solution of the quadratic equation

Maximum/Minimum - the highest or lowest point of a parabola, can be the vertex.

y-intercept - the point where the equation crosses the y-axis

x-intercept (zeros) - the point where the equation crosses the x-axis

end behavior - the behavior of the graph of f(x) as x approaches positive infinity or negative infinity.

Finding Zeroes

The zeroes of a function is the x-value that makes the function equal to zero.

Solve f(x) = 0

f(x) = -2x^2 - 5x + 7 = 0

Factor the expression -2x^2 - 6x + 8

(-2x - 7)(x - 1) = 0

and solve for x

x = -7 / 2 and x = 1

Graphing a Quadratic

To graph a quadratic, you need to find the y and x intercepts, and the vertex.

The vertex for this parabola would be (0,4). Graph the equation by plotting the vertex and the y-intercept as shown below. You may want to plot other points, also.

Transformations

Imaginary Numbers

A translation moves an object a distance in a given direction. The object remains the same size and shape, but changes location.

Ways to indicate that a translation will occur:

1. Description: 8 units to the left and 2 units down

2. Mapping: (x,y)-(x-7, y-3)

3. Notation: T(-7, -3)

4. Vectors: y= (-7,-3)

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i^2 = -1. The square of an imaginary number bi is -b^2. For example, 5i is an imaginary number, and its square is -25. When b^2-4ac= negative, you will have imaginary numbers.

Inequalities

Vertex Form

In order to solve a quadratic inequality, you have to graph the inequality first. Then you use the inequality symbol on the equation to determine the solutions. Use a dashed line for < or > and a solid line for < or >.

Shade below for less than symbols (<) and above for greater than symbols (>).

The vertex form of a quadratic function is given by

f (x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. To convert a quadratic from y = ax^2 + bx + c form to vertex form, y = a(x - h)^2+ k, you use the process of completing the square.

_ _

Equation in y = ax2 + bx + c form.

y = 2x^2 - 4x + 5

We will isolate the x^2 and x terms, move the + 5 to the other side of the equal sign.

y - 5 = 2x^2 - 4x

We need a leading coefficient of 1 for completing the square ... so factor out the current leading coefficient of 2.

y - 5 = 2(x^2 - 2x)

Create a perfect square trinomial. When we add a box to both sides, the box will be multiplied by 2 on both sides of the equal sign.

Find the perfect square trinomial. Take half of the coefficient of the x-term inside the parentheses, square it, and place it in the box.

Simplify and convert the right side to a squared expression.

y - 3 = 2(x - 1)^2

Isolate the y-term, so move the -3 to the other side of the equal sign.

y = 2(x - 1)2 + 3

Sometimes, you may need to transform the equation into the "exact" vertex form of y = a(x - h)2 + k, showing a subtraction sign in the parentheses before the h term, and the addition of the k term.

y = 2(x - 1)2 + 3

Vertex form of the equation.

Vertex = (h, k) = (1, 3)

(The vertex of this graph will be moved one unit to the right and three units up from (0,0), the vertex of its parent y = x2.)

Word Problem

Steve is hiking in the mountains. He wants to climb a ledge that's 20ft. above him. The height he throws the rope is given by the function h(t)=-16t^2+32t+2. What is the maximum height of the rope? Can he throw it high enough to reach the ledge?

To solve this problem, you must find the vertex. You can use the formula x=-b/2a to find the axis of symmetry.

Answer: No, Steve can only throw the rope a maximum of 18ft.

Chrislyn Kirkpatrick

3rd Period

Quadratics

t=-32/2(-16) = 1

Now you plug your t values into the equation t get your answer.

h(1)=-16(1)^2+32(1)+2

h(1)=-16+32+2 = 18ft.

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