Assignment -
Complete the following statements or answer the following questions:
1. For equilateral triangles, the three interior angle measurements are ________________.
2. Find the length of side and hypotenuse, given the length of the base = 1, for the 30-60-90 Triangle.
3. If a triangle is a right isosceles triangle, then the interior angle measurements are _______________.
4. Pythagorean Theorem: If a right triangle has two legs of length a and b with hypotenuse of length c, then_________________.
5. The distance between two points P(x1, y1) and Q(x2, y2) can be written as: d = ____________________.
6. The slope of a line passing through two points P(x1,y1) and Q(x2, y2) can be written as: m = ________.
7. How to find an equation of a line:
a) If you know the slope, m, and a point through which the line passes, then you can find the equation of this line.
- Write the equation of this line using the point-slope form.
- Write the equation of this line using the slope-Intercept form.
8. Interval Notation: if
(a, b) means a < x < b
_____ means a < x ≤ b
_____ means a ≤ x ≤ b
9. If a line has an equation of y = 3x + 2, then the point passing through could be written as: P(x, y) = (___,___).
10. When attempting to determine the domain of a function simply ask yourself what would break the function. What does not work? Give two examples for which the function does not exist.
11. The Principal Square Root: When you type in √4 into the calculator, the calculator will only give you the answer of 2. That is called the principal square root. That is the standard practice all around the world - even your calculator knows that. Thus if you were to be asked for the answer to √(x2), the appropriate answer is |x|. Thus you have found the principal square root. Now answer the following questions:
a. √(81)
b. If x2 = 81, then x = _____
12. Rewrite x-5 as a fraction: _______________
13. Rewrite the following fraction without any negative powers. Combine any like terms when possible:
((2x)-3yz5/3xy-4z-2) = _______
14. A function is increasing if as x gets bigger (which means that as you move from left to right along the graph) the y-values are increasing. A function is decreasing if as x gets bigger (which means that as you move from left to right along the graph, the y-values are decreasing.
a. Give an example of an increasing function: __________________
b. Give an example of a decreasing function: ___________________
c. Give an example of a constant function: _____________________
15. If you are solving an inequality such as x2 - 25 ≥ 0, just assume you are solving for the values for which x2 - 25 = 0; that will give you x = ±5. Now to find where x2 - 25 ≥ 0, put 5 and -5 on a number line and see if the values between them satisfy our condition, or if the values outside of 5 and -5 satisfy our condition. In this case it would be outside, so our solution set would be (-∞, 5] υ [5, ∞).
a. Find the solution to x2 - 16 < 0
b. Find the solution to -3x - 15 ≥ 0
16. In order to find the x and y value of the vertex point in a quadratic equation, all you have to do is to use this formula:
If f(x) = ax2 + bx + c, then ((-b/2a), f(-b/2a)) will be the vertex point.
When the quadratic equation is written in the graphing form: y = a(x - h)2 + k, then the vertex will be (h, k) where x = h, is the equation of the axis of symmetry. You can write the equation of a quadratic equation in this form by completing the square.
Find the vertex for the following quadratic equations:
a. y = 2(x-2)2 + 5
b. y = -3(x+3)2 - 3
c. y = x2 - 2x + 2
d. y = -2x2 - 4x - 7
17. Graph the following basic functions.
a. f(x) = √x
b. f(x) = |x|
c. f(x) = x2
d. f(x) = x3
e. f(x) = 2x
f. f(x) = log(x)
g. f(x) = 1/x
h. f(x) = 1/x2
i. f(x) = sin(x)
j. f(x) = cos(x)
k. f(x) = tan(x)
18. What is an asymptote? A vertical asymptote occurs when the x-value of that vertical line is one that is excluded from the domain of a function. Let's look at an example:
Suppose you have the function, f(x) = 2/(x-3). Note that x = 3 is x-value that is excluded from the domain. Therefore there is a vertical asymptote at x = 3. Now, also notice that it is impossible for f(x) to ever become 0. f(x) will have positive values and it will have negative values, but it cannot ever be equal to 0. In other words, 2/(x-3) ≠ 0. So there is going to be a horizontal asymptote at y = 0; y-values will get closer and closer to 0, but will never be actually equal to 0. Now...let's look at the graph of this function.
Find the Vertical Asymptote for f(x) = 2/(x+2) and then draw the basic graph.