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What is a Parabola?

Q: What are parabolas?

A: Parabolas are conic sections. The standard form is

The vertex is located at (p,q)

♦  Note that (x-p) means that the x-coordinate of the vertex is p ♦  Conversely, (x+p) means that the x-coordinate of the vertex is at -p

a is the ‘slope’ of the parabola, determining . . . → Read More: What is a Parabola?

Graphing a Parabola

Q: What if I don’t have an Equation Solver, how do I graph a parabola?

A: The easiest way is to make a table of values. To keep things simple, let’s look again at . To generate a table of values, simply substitute different values of x to obtain the corresponding value of y. . . . → Read More: Graphing a Parabola

Positive and Negative Parabolas

Q: What is the difference between a positive and a negative parabola?

A: The sign (+ or -) of the a value determines if the parabola opens upwards or downwards.

Let’s compare the graphs of , where a=+1 and , where a=-1. . . . → Read More: Positive and Negative Parabolas

Changing the value of a in Parabola equation y=a(x-p)^2+q

Q: What happens if I change the value of a?

A: This affects the steepness or ‘slope’ of the parabola.

Let’s compare 3 graphs with different a values: a=2, a=1, and a=0.5

Changing the value of p in Parabola equation y=a(x-p)^2+q

Q: What happens if I change the value of p?

A: p is the x-coordinate of your vertex. If it changes, the parabola will shift left or right.

Note that the general equation for a parabola is

(x-p) means:

if you have a positive p value, the result will be (x-p) if you . . . → Read More: Changing the value of p in Parabola equation y=a(x-p)^2+q

A Farmer Wants to Maximize Area

Q: A farmer wants to put a fence around a vegetable garden. Only three sides must be fenced, since a rock wall will form the fourth side. If he uses 40m of fencing what is the maximum area possible?

A:  Okay, so we have two widths and one length.

The material used would . . . → Read More: A Farmer Wants to Maximize Area

Max/Min Problem, Difference of Two numbers

Q: Two numbers hava a difference of 16. Find the numbers if their product is a minimum.

A: Let x = first number Let y = second number

Equation 1: x-y = 16 Equation 2: xy=P , where P is the product.

You need to isolate for x or y in Equation 1, and . . . → Read More: Max/Min Problem, Difference of Two numbers

Math 11: A Rectangle Word Problem using the Quadratic Formula

Formula for perimeter of a rectangle: P = 2w + 2l, where w=width and l=length

Formula for area of a rectangle: A = wl

30 = 2w+2l

40 =wl

In the area formula, isolate for l:

l = 40/w

Substitute this into the perimeter formula:

30 = 2w + 2(40/w) 30 = 2w + . . . → Read More: Math 11: A Rectangle Word Problem using the Quadratic Formula

Max/Min Rectangle Problem, Maximize the Area

Q: You have 128 feet of fencing to fence a rectangular area. Find the largest possible area to enclose this amount.

A: The perimeter of the space will be 128=2w+2L, where w=width and L=length

Isolate for L: L=64-w

The formula for the area of the rectangle is A=w*L. Substitute L=64-w into the area formula . . . → Read More: Max/Min Rectangle Problem, Maximize the Area

Max/Min Problem – Maximizing Revenue, Selling Calculators

Q: Calculators are sold to students for 20 dollars each. Three hundred students are willing to buy them at that price. For every 5 dollar increase in price, there are 30 fewer students willing to buy the calculator. What selling price will produce the maximum revenue and what will the maximum revenue be?

A: . . . → Read More: Max/Min Problem – Maximizing Revenue, Selling Calculators