1) Solve the following by factoring:

a)

Answer:
Show your work here:

b)

Answer:
Show your work here:

2) If , find

a) f(2)

Answer:
Show your work here:

b) f(-1)
Answer:
Show your work here:

3) Solve 6x2 + 3x – 18 = 0 using the quadratic formula.
Read the information in the assignment list to learn more about how to type math symbols, such as the square root.

Answer:
Show your work here:

4) Use the graph of y = x2 + 4x - 5 to answer the following:


a) Without solving the equation or factoring, determine the solution(s) to the equation, , using only the graph.

Answer:
Explain how you obtained your answer(s) by looking at the graph:

b) Does this function have a maximum or a minimum?

Answer:
Explain how you obtained your answer by looking at the graph:

c) What are the coordinates of the vertex in (x, y) form?
Answer:

d) What is the equation of the line of symmetry for this graph?

Answer:

5)
a) Calculate the value of the discriminant of .

Answer:
Show your work here:

b) By examining the sign of the discriminant in part a, how many x-intercepts would the graph of have? Why?

Answer:

6) a) Find the corresponding y values for x = -4, -3, -2, -1, 0, 1, 2 if .

Answer (fill in y column)
x y
- 4
- 3
- 2
- 1
0
1
2

Show your work here: (type x-squared as x^2 unless using a superscript feature).

b) Use Microsoft Excel to plot the points found in part a and to sketch the graph.
Read the information in the assignment list to learn more about how to graph in MS Excel.

Graph:

7) The path of a falling object is given by the function where represents the initial velocity in ft/sec and represents the initial height. The variable t is time in seconds, and s is the height of the object in feet.

a) If a rock is thrown upward with an initial velocity of 32 feet per second from the top of a 40-foot building, write the height equation using this information.

Typing hint: Type t-squared as t^2.

Answer:

b) How high is the rock after 0.5 seconds? Show all work.

Answer:
Show your work here:

c) After how many seconds will the rock reach maximum height? Show all work.

Answer:
Show your work here:

d) What is the maximum height? Show all work.

Answer:
Show your work here:

There is no question here

1) Solve the following by factoring:

a) To solve by factoring, we need to find two numbers that multiply to give the last term (-20) and add up to give the coefficient of the second term (-6). The numbers that satisfy these conditions are -10 and 4. So, we can rewrite the equation as (x - 10)(x + 4) = 0. Setting each factor equal to zero, we get x - 10 = 0 and x + 4 = 0. Solving for x, we find x = 10 and x = -4.

b) For this equation, we need to find two numbers that multiply to give the last term (12) and add up to give the coefficient of the second term (7). The numbers that satisfy these conditions are 3 and 4. So, we can rewrite the equation as (x + 4)(x + 3) = 0. Setting each factor equal to zero, we get x + 4 = 0 and x + 3 = 0. Solving for x, we find x = -4 and x = -3.

2) If f(x) = 3x^2 + 2x - 1, then:
a) To find f(2), we substitute 2 in place of x in the equation: f(2) = 3(2)^2 + 2(2) - 1. Simplifying, we get f(2) = 12 + 4 - 1 = 15.

b) To find f(-1), we substitute -1 in place of x in the equation: f(-1) = 3(-1)^2 + 2(-1) - 1. Simplifying, we get f(-1) = 3 + (-2) - 1 = 0.

3) To solve 6x^2 + 3x - 18 = 0 using the quadratic formula:
a) The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation. In this case, a = 6, b = 3, and c = -18. Plugging in these values, we get x = (-3 ± √(3^2 - 4*6*(-18))) / (2*6). Simplifying further, we have x = (-3 ± √(9 + 432)) / 12. Continuing to simplify, x = (-3 ± √441) / 12. This yields two solutions: x = (-3 + 21) / 12 and x = (-3 - 21) / 12, which simplify to x = 18 / 12 and x = -24 / 12. Therefore, x = 3/2 and x = -2.

4) Using the graph of y = x^2 + 4x - 5:
a) To determine the solution(s) to the equation x^2 + 4x - 5 = 0 using only the graph, we need to find the x-intercepts. These are the points where the graph crosses or touches the x-axis. By looking at the graph, we can see that the x-intercepts are approximately -5 and 1. Therefore, the solutions to the equation are x = -5 and x = 1.

b) To determine whether the function has a maximum or minimum, we can observe the shape of the graph. In this case, the graph opens upwards, which indicates a minimum. Therefore, the function has a minimum.

c) The vertex of a parabola in the form y = ax^2 + bx + c is given by the coordinates (-b/2a, f(-b/2a)). In this case, the vertex can be found using the formula x = -b/2a. Here, a = 1 and b = 4. Substituting these values, we get x = -4/2 = -2. To find the corresponding y-coordinate, we substitute x = -2 into the equation y = x^2 + 4x - 5: y = (-2)^2 + 4(-2) - 5 = 4 - 8 - 5 = -9. Therefore, the vertex is (-2, -9).

d) The equation of the line of symmetry for a parabola in the form y = ax^2 + bx + c can be found by using the formula x = -b/2a. In this case, the equation becomes x = -4/2 = -2. Therefore, the equation of the line of symmetry is x = -2.

5)
a) The discriminant of a quadratic equation in the form ax^2 + bx + c = 0 is given by the formula discriminant = b^2 - 4ac. In this case, a = 5, b = 12, and c = -3. Plugging in these values, we get discriminant = 12^2 - 4(5)(-3) = 144 + 60 = 204.

b) By examining the sign of the discriminant in part a (204), we can determine the number of x-intercepts the graph of the equation will have. If the discriminant is positive, the equation will have two distinct real x-intercepts. Therefore, the graph of the equation will have two x-intercepts.

6)
a) To find the corresponding y values for x = -4, -3, -2, -1, 0, 1, 2 in the equation y = x^2 + 2x + 1, we substitute each value of x into the equation and calculate y. The table would look as follows:

x y
-4 9
-3 4
-2 1
-1 0
0 1
1 4
2 9

b) To plot the points in Microsoft Excel, enter the x-values in one column and the corresponding y-values in another column. Then, select the data range and click on the "Insert" tab in the Excel ribbon. Choose the type of graph you want to create, such as a scatter plot or line graph. Excel will generate a graph based on the data points provided.

7)
a) The height equation for a falling object with an initial velocity of 32 feet per second from the top of a 40-foot building can be given by s(t) = -16t^2 + 32t + 40, where s represents the height of the object and t represents time in seconds.

b) To find the height of the rock after 0.5 seconds, we substitute t = 0.5 into the height equation: s(0.5) = -16(0.5)^2 + 32(0.5) + 40. Simplifying, we get s(0.5) = -4 + 16 + 40 = 52 feet.

c) To find the time at which the rock will reach maximum height, we can observe that the maximum height occurs at the vertex of the parabolic graph. Using the formula t = -b/2a, we can find the time. In this case, a = -16 and b = 32. Plugging these values, we get t = -32 / (2 * -16) = 1 second.

d) To find the maximum height, we substitute t = 1 into the height equation: s(1) = -16(1)^2 + 32(1) + 40. Simplifying, we get s(1) = -16 + 32 + 40 = 56 feet. Therefore, the maximum height of the rock is 56 feet.