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## PowerPoint Slideshow about 'Graphing E-portfolio' - ashanti

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Graphing E-portfolio

Quadratic Functions

Square Root Functions

Absolute Value Functions

Cubic Functions

Works Cited

Quadratic Functions

Definition: an equation in x that can be written in the standard form ax2 + bx+ c= 0, where a does not equal 0. The graph shape is a parabola or a capital letter U.

Domain: (-∞, ∞)

Range: [0, ∞)

How To Graph: Quadratic Functions

quadratic function equation: ax2 + bx + c =0.

- First, find the x-coordinate of the vertex by plugging in the b and a values into
- Second, make a table of values with the x-coordinate in the center and some x-values to the right and left.
- Then plug in the x-values into the equation to find the y-values.
- Plot the ordered pairs and connect them into a U-shaped parabola. ☺Remember: if a is positive, the parabola opens up; but if a is negative, the graph will open down.

Calculator Hints

- Press the Y= button and type in the quadratic equation.
- If any of the coefficients is a fraction, put brackets around the fractional number. The bracket buttons are located above the 8 and 9 number buttons.
- If the graph is too squished or too large, go to Tblset. Press the 2nd button and then the window button. The scale can be changed on ∆Tbl=. The start of the graph can be changed on TblStart=.
- The y-values of the graph can be found using the table of values found in the calculator. Press the 2nd button and then the Graph button to find the table of values.

Quadratic Word Problem

An object is launched at 19.6 meters per second (m/s) from a 58.8-meter tall platform. The equation for the object’s height at time t seconds after launch is s(t)= -4.9t2 + 19.6t + 58.8, where s is in meters.

Quadratic Predictions

- A tall and lovely giant named Thomas jumps up and intercepts the falling object at 9 meters. At what time would the object be caught?

The Solution Process

- Set s equal to 0
- To get rid of the nasty decimals, divide the equation by -4.9
- The new equation should be: 0 = t2 - 4t – 12. Find the x-coordinate of the vertex with the formula:
- Plug the x-coordinate into the equation to find the y-coordinate. The vertex is (2, -16).
- Make a table of values with five x-values to the left and five x-values to the right.
- Plug in the x-values into the equation to solve for the y-values.
- Graph the ordered pairs and look at the graph
- Find where y= 9

Word Problem Graph

y = 9 at -3 and 7, but the answer is 7 seconds because you can’t have -3 seconds.

x = -3 is an extraneous solution

Absolute Value Functions

Definition: an equation that shows the distance that a number x is from 0.

Domain: (-∞,∞)

Range: [0,∞)

How to Graph: Absolute Value Functions

absolute value equation:

- Find the x-coordinate of the vertex by solving for x in bx+c = 0. You have to set the terms in the absolute value brackets to zero to solve for x.
- Next, make a table of values with the x-coordinate of the vertex in the center. Include some x-values to the left and right of the vertex.
- Plug the x-values into the equation to find the y-values.
- Graph the points from the table of values and connect them in a V-shape. ☺Remember: if a is negative, the graph will open down. If a is positive the graph will open up.

Calculator Hints

- Press the Y=button.
- To make the equation an absolute value equation, press the Math button. Use the left arrow key to move over to Num. Press Enter on 1:abs(.
- Then type in the terms included in the absolute value brackets.
- If there’s an a value, type it in before the 1:abs(.
- If there’s a d value, be sure to close the absolute value brackets before typing it.

Cubic Functions

Definition: a continuous polynomial function with a highest degree of three

Domain: (-∞, ∞)

Range: (-∞, ∞)

How to Graph: Cubic Functions

a cubic function equation: a(x-h)3+k

- Find the vertex of the graph using (h,k). Set the values within the parentheses equal to zero and solve for h (the x-coordinate). K is the y-coordinate.
- Make a table of values with at least two x-values to both the left and right.
- Plug in the x-values into the equation to find the y-values. Try to find the zeros of the equation.
- Plot the points and connect them into a snake-shape. ☺Remember: A cubic function can have up to two turns. If the leading coefficient is positive, the graph will rise to the right and fall to the left. If it is negative, the graph will fall to the right and rise to the left.

Calculator Hints

- Press the Y= button and type in the equation. To make a third power, press the carrot(^) button.
- Another way to make a third power is to press the Math button and use the down arrow to scroll down to 3: 3.
- After typing the equation into the calculator, you can find the zeros using the table of values. Press the 2nd button and then the graph button.

Square Root Functions

Definition: the inverse function of the quadratic function y= x2 that finds the square root of a number x

Domain: [0,∞)

Range: [0, ∞)

How to Graph: Square Root Functions

square root equation:

- Find the origin of the graph using (h,k) or (a,b). Set the values within the square root stem equal to zero and solve for a (the x-coordinate). B is the y-coordinate.
- Make a table of values with the graph’s origin as the first point. Include x-values to the right of the origin only.
- Plot the ordered pairs and connect the points into a curve. ☺Remember: If the leading coefficient (the number before the square root stem) is negative then the curve will usually fall in the fourth quadrant, sometimes in the third. If the coefficient is positive, the curve will usually rise in the first quadrant, sometimes in the second.

Calculator Hints

- Press the Y= button to type in the square root function. To type in the square root stem, press the 2nd button and then the x2 button diagonal from the 7 button.
- If there’s a leading coefficient, type it in before the square root stem.
- Another way to type in a square root function is to raise the power to the one-half. Put the values that would normally be in the square root stem in parentheses. Press the carrot(^) button. Then, in parentheses, type 1/2.

Square Root Word Problem

- The Beaufort Scale represents the relationship between wind speeds and wind effects on a scale of 0 to 12. The scale can be modeled by the following square root function:

B represents the wind effects according to the points on the Beaufort Scale. X represents the speed of the wind in miles per hour. (pg. 383 of textbook)

0 Vertical Smoke

1 Light Air

2 Slight Breeze

3 Gentle Breeze

4 Moderate Breeze

5 Fresh Breeze

6 Strong Breeze

7 Moderate Gale

8 Fresh Gale

9 Strong Gale

10 Whole Gale

11 Storm

12 Hurricane

Beaufort Scale and Wind EffectsWord Problem Graph

Square Root Predictions

- One autumn afternoon winds reached 53 miles per hour. What was the wind effect and point of the Beaufort Scale on that day?

The Solution Process

- The origin of the graph does not need to be found because it’s a given that the graph begins at zero. The Beaufort Scale is from 0 to 12, only positive numbers. So the graph will only be in the first quadrant.
- Make a table of values starting with the ordered pair (0,0). Include numbers between 10 and 60. Because we want to find the scale point at 53 miles per hour, include 53 as an x-value.
- Plug in the x-values into the equation to find the y-values.
- Plot the ordered pairs and connect them into a curve.
- Find where x=53. Using the calculator, go to the table of values and find 53 (see Calculator Hints under square roots for help).

Word Problem Graph

At 53 miles per hour, the point on the scale is approximately 9.4. According to the Beaufort Scale, at 9.4 the wind conditions are between a strong and whole gale. Maybe a hurricane is going to come!!

Beaufort Scale

Works Cited

- Larson, Roland E., Timothy D. Kanold, and Lee Stiff. Algebra 2: an Integrated Approach. Evanston: McDougal Little, 1998.
- Stapel, Elizabeth. “Quadratic Word Problems: Projectile Motion.” Purplemath. 2006. 1 May 2006 <http://www.purplemath.com/modules/quadprob.htm>.
- Weisstein, Eric W. “Square Root.” Mathworld. 2006. Wolfram Web Resource. 10 May 2006 <http://mathworld.wolfram.com/SquareRoot.html>.

Rational- Zero Test

- To find the possible rational zeros of a cubic function, write down the p’s and the q’s. (p/q)
- Q is the leading coefficient and it goes to the denominator. P is the constant term of the function, or the term without a variable beside it. The p and its factors are listed from least to greatest in the numerator.
- Both p’s and q’s have values.
- Use synthetic division to find out which of the p’s are rational zeros of the function.
- The zeros are used as guidelines to graph a cubic function.

EX:

How To Graph: Cubic Functions

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