0. Write a paragraph describing in your own words what this unit was about and what you learned.
In this unit we learned what a quadratic formula is, a parabola which also included a vertex and the AOS witch is the axis of symmetry and more word problems.
1. How do you graph a parabola? Explain how to find the axis of symmetry, the vertex and how to get at least 5 points on your graph. How do you know if the graph will be opening upwards or downwards? How will you know if the graph has a maximum or a minimum and where will it be?
Include an equation and graph it out and explain how you use it to answer all of the questions above.
![Picture](/uploads/1/5/2/4/15247484/4377388.gif)
how you graph a parabola is by by finding the AOS, negative b over 2 times whatever a is. After finding the AOS, you must put that number into the equation, that will give you another number which is the y in, thus giving you the vertex. Depending on weather it is neg or pos, that will determine which if it is upward or downward.
2. How do you know how many roots a quadratic equation has by looking at the graph? Explain and include graphs.
![Picture](/uploads/1/5/2/4/15247484/9385422.gif)
After you have graphed the parabola the two points that are equally across from
each other, like -4 and 4, those will determine your two roots, as you see in
the picture
each other, like -4 and 4, those will determine your two roots, as you see in
the picture
3. How can you solve a quadratic by factoring and the zero product property? Why do you need to set the equation equal to zero to solve it? When the equations equals zero, how does this relate to the roots you can find by looking at the graph?
![Picture](/uploads/1/5/2/4/15247484/892181.jpg)
Here is an example of solving a quadratic formula by factoring.
x^2 + 12x +
35 = 0
(x+7) (x+5)
x+7=0 x+5=0
-7 -7 -5 -5
x=-7
x= -5
You need to set it up this way because you need to make sure that the equation is true.
x^2 + 12x +
35 = 0
(x+7) (x+5)
x+7=0 x+5=0
-7 -7 -5 -5
x=-7
x= -5
You need to set it up this way because you need to make sure that the equation is true.
4. What is the quadratic formula and how do you use it? Include a quadratic equation and show how to solve it using the quadratic formula.
![Picture](/uploads/1/5/2/4/15247484/4095512.jpg)
Step 1: 3 squared = 9 and than you do 4 x 1 x -4 which = 16 and than one the
bottom do 2 x 1 which is 2.
Step 2: 9 + 16 which is 25.
Step 3: After you
square root 25 which is 5.
Step 4: -3 = and - 5 over 2
Step 5: -3 +5 over 2 and than -3 -5 over
Step 6: and than you solve
bottom do 2 x 1 which is 2.
Step 2: 9 + 16 which is 25.
Step 3: After you
square root 25 which is 5.
Step 4: -3 = and - 5 over 2
Step 5: -3 +5 over 2 and than -3 -5 over
Step 6: and than you solve
5. When using the quadratic formula? What does it mean if the discriminant is
negative? What does it mean if the discriminant is positive? What if it is zero? Many solutions will it have? Explain why and relate it back to the graphing and finding roots.(see page 471 for details)
When using the quadratic formula to solve a quadratic equation (ax2 + bx + c =
0), the discriminant is b2 - 4ac. This discriminant can be positive, zero, or
negative. Create three unique equations where the discriminant is positive,
zero, or negative. For each case, explain what this value means to the graph of y = ax2 + bx + c. It will have either have none, one or two.
0), the discriminant is b2 - 4ac. This discriminant can be positive, zero, or
negative. Create three unique equations where the discriminant is positive,
zero, or negative. For each case, explain what this value means to the graph of y = ax2 + bx + c. It will have either have none, one or two.
6. What kinds of word problems can be solved using Quadratics? Find one in chapter 9 of your textbook and solve it. (Note page 463-468 has many good examples)
A motorboat makes a round trip on a river 56 miles upstream and 56 miles
downstream, maintaining the constant speed 15 miles per hour relative to the
water.
The entire trip up and back takes 7.5 hours.
What is the speed of
the current?
Solution
Denote the
unknown current speed of the river as miles/hour.
When motorboat moves
upstream, its speed relative to the bank of the river is miles/hour, and the
time spent moving upstream is hours.
When motorboat moves downstream, its
speed relative to the bank of the river is miles/hour, and the time spent
moving downstream is hours.
So, the total time up and back is , and it is
equal to 7.5 hours, according to the problem input.
This gives an equation .
To simplify the equation, multiply both sides by and collect common terms.
Step by step, you get
Answer. The speed of the current is 1
mile/hour
downstream, maintaining the constant speed 15 miles per hour relative to the
water.
The entire trip up and back takes 7.5 hours.
What is the speed of
the current?
Solution
Denote the
unknown current speed of the river as miles/hour.
When motorboat moves
upstream, its speed relative to the bank of the river is miles/hour, and the
time spent moving upstream is hours.
When motorboat moves downstream, its
speed relative to the bank of the river is miles/hour, and the time spent
moving downstream is hours.
So, the total time up and back is , and it is
equal to 7.5 hours, according to the problem input.
This gives an equation .
To simplify the equation, multiply both sides by and collect common terms.
Step by step, you get
Answer. The speed of the current is 1
mile/hour