Flashcards in 1.2.3 Parabolas Deck (15):

1

## Parabolas

###
• The graph of a second-degree polynomial expression is a parabola. A parabola consists of the set of all points in a plane that are equidistant from a fixed line (the directrix) and a fixed point not on the line (the focus).

• When graphing functions, start by looking for ways to simplify their expressions. Always promise that the denominator will not equal zero when you cancel.

• The distance formula is an application of the Pythagorean theorem. d = square root (x2-x1)^2 + (y2-y1)^2

2

## note

###
- In general, a parabola can be expressed by

f(x) = ax2 + bx + c. Each of the constants a, b, and c has a

different effect on the appearance of the parabola.

- If the coefficient of the x-squared term is positive, the

parabola will open upwards. You can think of it as a

happy-faced parabola. If the coefficient of x-squared is

negative, the parabola will open downwards as a sad-faced parabola.

- If the coefficient of the x-squared term is greater than 1 or less than –1, the parabola will be stretched vertically, making it look thinner and tighter.

- Conversely, if the coefficient of the x-squared term is between 0 and 1 or between 0 and –1, the parabola will be compressed vertically, making it look wider.

- The presence of the constant c has the effect of shifting the parabola vertically. For the function on the far left, the constant 1 moves the parabola upwards one unit. A negative constant would move the parabola downwards.

- Changing the value of the coefficient of x has the effect of moving the parabola around in a manner that is a bit more difficult to predict.

- Here is a function expression that does not resemble the general form for a parabola.

- Notice that you can factor x from the numerator. When you cancel it with the x in the denominator, you must agree not to evaluate the function at x = 0.

- The expression that results from the cancellation is that of a happy-faced parabola shifted up one unit. Make sure to leave a hole in the graph at x = 0. The function is not defined at that point.

- To determine the distance between the two points (x1

, y1) and (x2, y2), use the Pythagorean theorem. By connecting the points with a line segment you can construct a right triangle whose legs are parallel to the x- and y-axes. The lengths of the legs are given by x2 – x1 and y2 – y1 . You can then solve for the length of the hypotenuse d, which is the distance between the two points.

- The formula you produce is called the distance formula. You can either memorize the formula or use the Pythagorean theorem to derive it when you need it.

3

##
Does the parabola described by the

function

f(x)= -2(x^2+7) -4x^2 -9(5-x^2) open upwards or downwards?

### upwards

4

## A line intersects a parabola at the points (−2, 3) and (4, 11). What is the distance between the two points of intersection?

### d = 10

5

## Which of the following is the quadratic function whose graph is the parabola shown?

### f(x) = x^2 -2x+2

6

## Amanda and Laura are in the middle of a hiking trip and had a disagreement as to which direction to travel. Laura decides to hike due east in search of civilization and Amanda begins moving due south. In two hours Laura has moved 4 miles and Amanda has moved 5 miles. How far apart are the hikers at this time?

### d = √41

7

##
What is the y-intercept of the graph of

f(x) = -(x-2)^2 +3x +1

### -3

8

##
Does the parabola described by the

function

f(x) = 3(x^2-2)-6x^2-(2+x^2)

open upwards or downwards?

### downwards

9

## What is the distance between the two points depicted in this graph?

###
d=√52

10

## Which of the following is the graph of the parabola y = −2x ^2 + 4x ?

###
right side of y-axis, opens down

(0,0), (1,2), (2,0)

11

## What is the distance between the two points (−1, 4) and (2, 5)?

### √10

12

##
Find the distance between the two points

(-2,5) and (-3,7)

### √5

13

## Which of the following is the quadratic function whose graph is the parabola shown?

###
None of the above

(actual answer -4x^2-8x-4)

14

## What is the distance between the two points (1, 5) and (−3, 10)?

### √41

15