"Quadratic Functions" Re-Test
By Courtney Little
x=-.25 (-1/4) x=2.5 (5/2)
- To find the factors you enter the quadratic formula into the calculator. Find both zeros, which for this equations is (-1/4) and (5/2). To turn zeros into factors you need to get both the x and the zero on the same side. For fractions the denominator or the bottom number to moved in front of the x, (4x+1) and (2x-5). Don't forget to switch the sign of the zeros when turned into factors.
- To find the factors for this one you again enter it in to the in the y= of the calculator. Find both zeros, which would be x=3 and x=-4. To get the factors you need to get the x and the zero on both sides, so subtract 3 for x=3 and add four for x=-4. Therefore the factors you get are (x-3) and (x+4).
Using Square Root Method:
-We need to get the numbers on one side and the x on the other. So add one to both sides. Now that both the numbers are on one side and the x is on the other we need to get rid of the squared. So square the x^2, and what you do to one side you need to do to the other, so square root 25 too. You end up with x=5.
-We need to get the numbers on one side and the x on the other, so we subtract 28 from both sides. Now we need to get the 3 over, the opposite of multiplication is division. So we divide both sides by 3. Then square root both sides. The number inside the square root is a negative, so when we look for pairs to come out of the square root an i also comes out. The i comes out in place of the negative. We end of with a pair of 3's, so a 3 and an i come out. Our ending answer is x=+-3i.
-This problem we will use the discriminant so remember the standard ax^2+bx+c=0. To find the discriminant we use the formula b^2-4(a)(c). So square the -5, but remember to put parenthesis around -5. The discriminant is 49, so now we will put it in -b+-√b^2-4ac
/2. The b value becomes positive since it is already negative. we know the discriminant is 49, so it goes into the square root. Then we multiply the a value by 2. When we square root the 49, we get a pair of sevens, so a seven comes out. Now add 5 plus 7 divided by 2, and 5 minus 7 divided 2. 6 and -1 are our answers.
-Find the discriminant, using (b)^2-4ac, which we find to be -4. So when we square root -4, we get a pair of two's and an i for the negative. So the equation becomes -2+-2i
/2. We can divide the two out because they are all divisible by 2. So we should end up with -1+-i.
Completing the Square Root
-The first thing we do is divide everything by -2. Then set the equation equal to zero. Add 3 to both sides to get on the right side of the equal sign. Divide 2 by 2 and square it, to get 1. So 1 is what we add to both sides. we pull the x down and the first sign, which is a negative. The square root of 1 is 1, so we get (x-1)^2=4. Now we square root both sides, so the squared sign goes away and becomes x-1=+-√4. We get a pair of two's, so a 2 comes out. Now add the 1 to both sides to get x=1+-2. We now do 1+2, and 1-2, to get our answers 3 and -1.
-There is not a number in front of the x, so we start by setting the equation equal to zero and subtracting 14 from both sides. divide 6 by 2 and square it, and we find we are adding 9 to both sides. Now square root both sides to get rid of the square. Subtract 3 from both sides to get it on the right side. It goes in front of the +- signs in front of the square root. So an i comes out, but 5 can't because it can not be broken down any smaller so it stays in the square root and we end up with x=-3+-i√5.
- Set it up like the formula, (30)^2-4(9)(25)=0
-The zero means that it only touches the x-axis once.
-If the equation was entered into the y= in the calculator you would see that the graph would only touch once.
Two Real Solution: 4x^2-6x-7
- Set it up like the formula , (-6)^2-4(4)(-7)=148
-The number means that it touches the x-axis twice.
Two Imaginary Solution: 2x^2+6x+7
- Set it up like the formula, (6)^2-4(2)(7)=-20
-The negative number means that the graph doesn't touch the x-axis at all
Turnig Standard Form into Vertex Form
-Since there is no number in front of x, we will start by setting the equation equal to zero. Subtract 13 from both sides. Now divide the 6 by 2 and square it to get 9, and that is what we will add to both sides. Add 4 to both sides and we should end up with a vertex equation f(x)=(x+3)^2+4
-Since there is a number in front of the x we will divide everything by -3. Now we set the equation equal to zero and add 1 to both sides. We will subtract the 5 from both sides. The -3 that we divided everything by, is brought back into the vertex form as the d value which is in front of the equation. The vertex form we get is f(x)=-3(x-2)^2-5.
- The vertical shift is the the number (Y)on the outside that shift the graph up and down.
Horizontal Shift: (x+3)^-4, (x-3)^2-9
-The horizontal shift is the number in the parenthesis (X) that shift the graph left or right.
Vertical Stretch: 2(x+4)^2-7
-The vertical stretch is what makes the graph narrower, and it is found in the d value at the front of the equation.
Horizontal Stretch: 1/2(x-4)^2+9
-The horizontal stretch is what makes the graph wider or fatter, and is found in the d value at the front of the equation.
Reflections over the x-axis: -(x-3)^2+3
-The reflections over the x-axis is when you have a negative in the d value
The vertical stretch makes the graph more narrow, the blue dots are the parent stretch.
The graph shifts up and down with a vertical shift
The graph shifts left and right with a horizontal shift
A horizontal stretch stretches the graph, makes it fatter. The blue dots are showing a normal parent stretch.
Reflection over the x-axis
With a - in the d value the graph faces down reflecting over the x-axis.
Types of Answer
Roots: Can be Real or Imaginary numbers, this means that the answer can have a negative number in the square root and end with an i.
Zeros:Can only have Real numbers, with zeros you can actually go and look at it on the calculator, so it can not have a negative in the square root or end with an i. If it does then it will be no solution.
Ex: x=5+-√-503 =No Solution
X-Intercepts:Can be Real numbers only
Ex:x=-5+-√49 =(1/3), -2
Solutions:Can be Real or Imaginary numbers