Please Help!
Given the following three points, find by hand the quadratic function they represent.
(−1,−8), (0,−1),(1,2)
A. f(x)=−3x2+4x−1
B. f(x)=−2x2+5x−1
C. f(x)=−3x2+10x−1
D. f(x)=−5x2+8x−1
Determine if the following set of ordered pairs represents a quadratic function. Explain.
(5, 7), (7, 11), (9, 14), (11, 18)
A. The y-values go up by the square of the x-value (22=4). Therefore, the ordered pairs represent a quadratic equation.
B. The y-values go up by the square of the x-value (22=4). Therefore, the ordered pairs do not represent a quadratic equation.
C. Since the differences between the x-values is 2 and the differences between the y-values is 4, that means that the differences between the differences of the y-values are all zero. Therefore, the ordered pairs represent a quadratic equation.
D. Since the differences between the differences of the y-values is not consistent, the ordered pairs do not represent a quadratic equation.
To find the quadratic function represented by the given three points, we can start by assuming the function has the form f(x) = ax^2 + bx + c. Then, we can use the three given points to construct a system of equations to solve for the coefficients a, b, and c.
Using the first point (-1, -8):
-8 = a(-1)^2 + b(-1) + c
-8 = a - b + c
Using the second point (0, -1):
-1 = a(0)^2 + b(0) + c
-1 = c
Using the third point (1, 2):
2 = a(1)^2 + b(1) + c
2 = a + b + c
Now, substituting c = -1 into the first and third equations:
-8 = a - b - 1
2 = a + b - 1
Simplifying the equations:
a - b = -7 [Equation 1]
a + b = 3 [Equation 2]
Adding Equation 1 and Equation 2, we get:
2a = -4
a = -2
Substituting a = -2 into Equation 2:
-2 + b = 3
b = 5
Finally, substituting a = -2 and b = 5 into c = -1, we get:
c = -1
Therefore, the quadratic function represented by the three points is:
f(x) = -2x^2 + 5x - 1
Hence, the answer is B. f(x) = -2x^2 + 5x - 1.
For the second question, to determine if the set of ordered pairs represents a quadratic function, we need to check if the differences between the y-values increase by the square of the x-value for each consecutive pair.
Calculating the differences:
7 - 5 = 2
11 - 7 = 4
14 - 11 = 3
18 - 14 = 4
As we can see, the differences between the y-values are not increasing by the square of the x-values. Therefore, the set of ordered pairs does not represent a quadratic function.
Hence, the answer is B. The y-values go up by the square of the x-value (2^2 = 4). Therefore, the ordered pairs do not represent a quadratic equation.
To find the quadratic function represented by the given three points (−1,−8), (0,−1), (1,2), we can use the formula for a quadratic function: f(x) = ax^2 + bx + c.
First, substitute the x and y values from each point into the equation f(x) = ax^2 + bx + c:
For the first point (-1, -8):
-8 = a(-1)^2 + b(-1) + c (equation 1)
For the second point (0, -1):
-1 = a(0)^2 + b(0) + c (equation 2)
For the third point (1, 2):
2 = a(1)^2 + b(1) + c (equation 3)
We now have a system of three equations with three unknowns (a, b, c). To solve this system, we can use various methods such as substitution or elimination.
A common method is substitution. Let's rewrite equation 2 and substitute it into equations 1 and 3. Equation 2 becomes:
-1 = c (since a(0)^2 and b(0) are both zero)
Now, substitute -1 for c in equations 1 and 3:
Equation 1: -8 = a(-1)^2 + b(-1) - 1
-8 = a - b - 1
Equation 3: 2 = a(1)^2 + b(1) - 1
2 = a + b - 1
We now have a system of two equations with two unknowns (a, b):
-8 = a - b - 1 (equation 4)
2 = a + b - 1 (equation 5)
To solve this system, we can add equations 4 and 5:
-8 + 2 = (a - b - 1) + (a + b - 1)
-6 = 2a - 2
To solve for a, divide both sides by 2:
-3 = a - 1
Now, add 1 to both sides:
-2 = a
Now that we know a = -2, we can substitute this value back into equation 5 to solve for b:
2 = (-2) + b - 1
Simplify the equation:
2 = -2 + b - 1
Combine like terms:
2 = -3 + b
To solve for b, add 3 to both sides:
5 = b
Now that we know a = -2 and b = 5, we can substitute these values back into one of the original equations to solve for c. Let's use equation 2, since c was already determined to be -1:
-1 = a(0)^2 + b(0) + c
-1 = 0 + 0 + c
Therefore, c = -1.
Now we have determined all the coefficients of the quadratic function f(x) = ax^2 + bx + c:
f(x) = -2x^2 + 5x - 1
Therefore, the quadratic function represented by the given three points is option B: f(x) = −2x^2 + 5x − 1.
To determine if the set of ordered pairs (5, 7), (7, 11), (9, 14), (11, 18) represents a quadratic function, we need to check if the differences between the y-values are consistent and proportional to the differences between the x-values.
Looking at the differences in the y-values: 7-11 = -4, 11-14 = -3, 14-18 = -4.
The differences in the y-values are not consistent. Therefore, the set of ordered pairs does not represent a quadratic function.
Therefore, the correct answer is option D: Since the differences between the differences of the y-values are not consistent, the ordered pairs do not represent a quadratic equation.
let the function be y = ax^2 + bx + c
for (0,-1) , -1 = 0 + 0 + c , so c = -1
and our equation is y = ax^2 +bx - 1
for (-1,-8)
-8 = a - b - 1 ----> a - b = -7
for (1,2)
2 = a + b - 1 ---> a + b = 3
add them
2a = -4
a = -2
in a+b=3 , -2+b = 3
b = 5
so which choice matches my answer?
do the points
, (11, 18) represent a function?
I don't know what properties you learned, but I would do it this way,
x -- y --1st diff -- 2nd diff
5 7
7 11 -- 4
9 14 -- 3 --- -1
11 18 -- 4 --- +1
The second differences should be a constant, they are not, so .... no quadratic
another way would be to find the quadratic using the first 3 points, then testing
if the 4th given point satisfies this equation, it does not, so NO quadratic
first one only, you do the rest
note the middle point (0,-1) works for all the equations, y = -1 for any x
now the first point (-1 , -8)
A , -3 - 4 - 1 = -8 ok so far
B , -2 - 5 - 1 = -8 ok so far
C , -3 -10 -1 =-14 no way
D , -5 - 8 - 1 = -14 no way
so it is A or B
try last point (1,2)
A gives -3 + 4 - 1 = 0
B gives -2 + 5 -1 = 2 !!!!! yes, it is B