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When a ball is thrown vertically upward, its height (h, in feet) above the ground after t seconds is described by the mathematical formula h = - 16t2 + 96t + 80. At what times is the ball 103 feet above the ground?



Use the quadratic formula:
a=16, b=-96, c=23



and
and
and
x=.25 s and x=5.75 s
One the way up, the ball hits 103 ft in .25 s, on the way back down it hits it again in 5.75 s.
Since this came out to rational answers this could have been factored. When the numbers are more difficult to deal with the quadratic formula is easier to use than factoring. Here's what it looks like factored.

4t-1=0 and 4t-23=0
4t=1 and 4t=23
t=1/4 s and t=23/4 s
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1st factor the denominators.

The LCD is (x-1)(x-5)(x-6), whatever you do to the denominator you must also do to the numerator.




You're right!
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Let f(x)=x+5; let g(x)=X^2-2; let h(x)=-3x
1. ((f)*(g))(x) <---I think this is what you meant.




:
2. h(h(5))
First find h(5)


h(h(5))=h(-15)
Now substitute that value into the h function.
h(h(5))=-3(-15)
h(h(5))=45
:
3. g(h(-1))
h(-1)=-3(-1)
h(-1)=3
g(h(-1))=g(3) Substitute 3 into the g function.

g(h(-1))=9-2
g(h(-1))=7
:
4. (f(4)) + (g(6))

(4+5)+(36-2)
9+34
43
:
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4x-5=+-
4x=5+-
and
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The slope intercept form of a line is , m=slope, and (0,b) is the y-intercept.
Find the slope and y-intercept.
–x – 4y = –32
x-x-4y=x-32
-4y=x-32


, the slope m=-1/4
, the y-intercept is (0,b)=(0,8), some teachers are happy with 8.
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The slope intercept form of a line is , m=slope and (0,b)=y-intercept.
Write the equation of the line with slope 4 and y-intercept (0, –5). Then graph the line.
m=4 b=-5

To graph, plot the y-intercept (0,-5)
From that point use the slope to graph another point.
m=4=4/1. Rise (go up) 4 and run (go right) 1, you should be at (1,-1). Plot another point there. Connect the points and you have a line:

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The slope intercept form of a line is , m=slope, and (0,b) is the y-intercept.
Find the slope and y-intercept.
–x – 4y = –32
x-x-4y=x-32
-4y=x-32


, the slope m=-1/4
, the y-intercept is (0,b)=(0,8), some teachers are happy with 8.
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how can i solve the system by graphing.
E1: x + y = 4
E2: –x + y = 2
:
For E1, let x=0 and y=?
0+y=4
y=4 Plot (0,4)
Now let y=0 and x=?
x+0=4
x=4 Plot (4,0)
Connect your points and you have this line:

For E2, let x=0 and y=?
-0+y=2
y=2 Plot (0,2)
Now let y=0 and x=?
-x+0=2
-x/-1=2/-1
x=-2 Plot (-2,0)
Connect these points and you have a second line.

The solution is their point of intersection: (1,3)
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Find the coordinates of the vertex of

When the quadratic formula is in standard form: the x-coordinate is found with the formula:
a=2, b=0, and c=0



That's the x-coordinate of the vertex. You find the y coordinate by letting x=0.


y=0
The vertex is (x,y)=(0,0)
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i have to find the coordinates of the vertex of y=-7x^2+2x
The x coordinate of the vertex is found with the formula:
The quadratic formula is in standard form:
a=-7, b=2, and c=0


The y coordinate is what you get when you substitute the x coordinate into the equation and solve for y.




The vertex is (-1/7,-3/7).
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Type in your problem and directions and we'll help you. We don't have your book.

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To divide 7/x+2 by x-2/14, you can multiply 7/x+2 by what?
When dividing fractions, you flip the fraction over that's behind the division symbol and multiply. It's called multiplying by the reciprocal.
The reciprocal of is
So you multiply by
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Hi ac,
The length of a rectangle is 5 cm more than 2 times its width. If the area of the rectangle is 78cm^2, find the dimensions of the rectangle to the nearest thousandth.
:
Let the width be: x
Then the length is: 2x+5
Area is: 78 cm^2
Area=length*width
:
78=x(2x+5)
78=2x^2+5x
0=2x^2+5x-78
This is in standard form:
We can use the quadratic formula to solve it:
a=2, b=5, and c=-78



You can ignore the negative dimension.
x=5.119
The width is: x=5.119 cm
Length is:2x+5=2(5.119)+5=15.238 cm
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Hi CH
find the x-intercepts??
y=x^2+6x+8 and
Let y=0 and solve for x.
0=x^2+6x+8 Factor
0=(x___)(x___) the blank gets filled with two numbers that multiply to get 8 and add to get 6. 2*4=8 and 2+4=6
0=(x+2)(x+4) Let each parentheses=0
x+2=0 and x+4=0
x=-2 and x=-4
The x-intercepts are (-2,0) and (-4,0)
:
y=x^2-2x+4
This has no x-intercepts. It's not factorable. If you're familiar with the discriminant:, you can tell if a quadratic equation has no x-intercepts if the discriminant is negative.
There are no x-intercepts.
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Rewrite the equation 4x – 10y = 11 as a function of x.


-4x+4x-10y=-4x+11
-10y=-4x+11

<--some books are happy here
<--others want the y to be written in function notation.
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Given f(x) = –2x + 4, find f(a – 2).
Replace x with (a-2) in f(x):
f(a-2)=-2(a-2)+4
f(a-2)=-2(a)-2(-2)+4
f(a-2)=-2a+4+4
f(a-2)=-2a+8
:
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Solve the system by addition.
Add the two equations and the y's are eliminated. Now you can solve for x.
x + y = 4
x – y = 10
_____________
2x+0=14
2x=14
2x/2=14/2
x=7
:
Let x=7 in either equation and solve for y. I'm using the first one.
7+y=4
-7+7+y=4-7
y=-3
:
The solution is (x,y)=(7,-3)
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How does the graph of the quadratic function or the absolute value function change if you subtract something from them?
They cause a vertical shift down that many units.
y=x^2 and y=x^2-3 are graphed below the -3 caused a vertical shift down 3 units

Y=|x| and y=|x|-3 are graphed below and the -3 caused the same vertical shift down three units.

:
When the 3 is subtracted from the x it causes a horizontal shift to the right three units.
y=x^2 and y=(x-3)^2 are graphed below.

y=|x| and y=|x-3| are graphed below.

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graph each system of equations and describe it as consistent and independent,consistent and dependent, or inconsistent.
y=-x-2
x+y=-4
:
For y=-x-2
Let x=0 and y=?
y=-(0)-2
y=-2 Plot (0,-2)
Now let x=-2 and y=?
y=-(-2)-2
y=2-2
y=0 Plot (-2,0)
Connect the points and you have this line.

For x+y=-4
Let x=0 and y=?
(0)+y=-4
y=-4 Plot (0,-4)
Let y=0 and x=?
x+(0)=-4
x=-4 Plot (-4,0)
Connect the points and you have a second line.

Notice that the lines a parallel and will never intersect.
This system has no solution and is inconsistent (and independent).
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(a-2)(a-18)=0
a-2=0 and a-18=0
a=2 and a=18 Check for extraneous solutions:



a=2 is a valid solution.



a=18 is extraneous.
The only valid solution is a=2.
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Solve:
(a+b)^3
Most people end up having to memorize this, this is how you derive it.
(a+b)(a+b)(a+b) Multiply the second two.
(a+b)(a^2+ab+ab+b^2)
(a+b)(a^2+2ab+b^2) distribute the a then the b from the first parentheses.
a(a^2+2ab+b^2)+b(a^2+2ab+b^2)
a^3+2a^2b+ab^2+a^2b+2ab^2+b^3
a^3+2a^2b+a^2b+ab^2+2ab^2+b^3
a^3+3a^2b+3ab^2+b^3
If you're in algebra II it's worth memorizing.
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use the reference angles to find the exact value of the expression:
This falls in Quad. III, sine and its reciprocal is negative there.
csc(=
=
=
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factor completely.

2b*b=2b^2 and 1*1=1 and 2b*1+b*1=3b, so

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multiplying polynomials





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6x^2-13x+6=0
We need to factor the left side which is in standard form:
a=6, b=-13, and c=6
Replace the middle term with two numbers that multiply to give you a*c=6*6=26, but add to give you b=-13. -4*-9=36 and -4+-9=-13
6x^2-4x-9x+6=0 Factor by grouping
(6x^2-4x)+(-9x+6)=0 group
2x(3x-2)-3(3x-2)=0 factor out the GCF in each group
(3x-2)(2x-3)=0 Factor the 3x-2 out of each.
3x-2=0 and 2x-3=0
3x=2 and 2x=3
3x/3=2/3 and 2x/2=3/2
x=2/3 and x=3/2
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how do you solve a problem like this: 3m=(-84)
Notice that this means 3 times a number = -84.
To isolate the m we need to do the opposite of 3 times on both sides of the equation. We divide both sides of the equation byt 3.



:
Sanity Check, do both sides of the = sign equal when m=-28?
3(-28)=-84
-84=-84 We're right.
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What would a x value, that is undefined, be for the equation y=2/(1-2x)?
The fraction is undefined when the denominator=0
1-2x=0
-2x=-1
-2x/-2=-1/-2
x=1/2 The equation is undefined when x=1/2
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12/16 times 4/3 divided by 5/6
divided by





:
If your observant it's faster to reduce first.





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Graph each function
y=x^2-5x+6 <--you can use ^ to raise something to a power.
This is a parabola.
Because x^2 is positive, we know it opens up.
The y-intercept can be found by letting x=0
y=0^2-5(0)+6
y=0+0+6
y=6 Plot (0,6)
:
To find the x-intercept, let y=0
0=x^2-5x+6
0=(x-3)(x-2)
x-3=0 and x-2=0
x=3 and x=2
Plot (3,0) and (2,0)
The axis of symmetry and x coordinate of the vertex can be found using the formula
a=1, b=-5, and c=6


x=2.5

y=6.25-12.5+6
y=-.25 Plot (2.5,-.25)
You now know that it looks like this.

:
f(x)=x^2+4
x^2 is positive, so it opens up.
The y-intercept is:
f(0)=0^2+4
f(0)=4 Plot (0,4)
:
There is no x-intercept because 0=x^2+4 has no real solution.
:
The vertex is the y intercept because there's no b. Because you only know one point. Plot some points on the left and right side of x=0.
f(1)=1^2+4
f(1)=1+4
f(1)=5 Plot (1,5)
f(-1)=(-1)^2+4
f(-1)=1+4
f(-1)=5 Plot(-1,5)
I'll let you find and plot f(-2) and f(2).
Connect your points and you have:

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In the equation x/2 + y/3 = 1, how do you find the x and y intercepts?
Let x=0 and solve for y to get the y-intercept.




The y-intercept is (0,3)
Let y=0 and solve for x to get the x-intercept.




The x-intercept is (2,0)
Here's what it looks like graphically, notice the line crosses the y axis at (0,3) and the x axis at (2,0).

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