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AP Calculus AB: 1.2.3 Parabolas

Mathematics15 CardsCreated 3 months ago

This content introduces parabolas as the graphs of second-degree polynomial functions, explaining how coefficients affect their direction, width, and position. It also covers simplifying expressions, handling undefined points, and using the distance formula derived from the Pythagorean theorem to calculate distances between points.

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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

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Key Terms

Term
Definition

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 equidist...

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 appea...

Does the parabola described by the

function

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

upwards

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

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

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

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

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AP Calculus AB: 1.2.3 Parabolas
TermDefinition

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

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.

Does the parabola described by the

function

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

upwards

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

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

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

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

What is the y-intercept of the graph of

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

-3

Does the parabola described by the
function
f(x) = 3(x^2-2)-6x^2-(2+x^2)
open upwards or downwards?

downwards

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

d=√52

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

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

√10

Find the distance between the two points

-2,5) and (-3,7

√5

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

None of the above

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

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

√41

Does the parabola described by the function
f(x) = -(1-4x^2)+7x^2-(4-10x^2)
open upwards or downwards?

Upwards