Which polynomial function has x intercepts -1,0, and 2 and passes through the point (1,-6)?
1 answer:
Answer:
f(x) =
Step-by-step explanation:
Which polynomial function has x intercepts -1,0, and 2 and passes through the point (1,-6)?
There are 3 distinct and real roots given in the question, which means that the function must be a third degree polynomial. The roots are -1, 0, and 2. This means that f(x) = 0 at these points. The general form of the cubic equation is given by:
f(x) = ax^3 + bx^2 + cx + d; where a, b, c, and d are arbitrary constants.
From the given data:
f(-1)=0 implies a*(-1)^3 + b*(-1)^2 + c(-1) + d = -a + b - c + d = 0. (Equation 1).
f(0)=0 implies a*(0)^3 + b*(0)^2 + c(0) + d = 0a + 0b + 0c + d = 0. (Equation 2).
f(2)=0 implies a*(2)^3 + b*(2)^2 + c(2) + d = 8a + 4b + 2c + d = 0. (Equation 3).
f(1)=0 implies a*(1)^3 + b*(1)^2 + c(1) + d = a + b + c + d = -6. (Equation 4).
Equation 2 shows that d = 0. So rest of the equations become:
-a + b - c = 0;
8a + 4b + 2c = 0; (Divide 2 on both sides of the equation to simplify).
a + b + c = -6
This system of equation can be solved using the Gaussian Elimination Method. Converting the system into the augmented matrix form:
• 1 1 1 | -6
• -1 1 -1 | 0
• 4 2 1 | 0
Adding row 1 into row 3:
• 1 1 1 | -6
• 0 2 0 | -6
• 4 2 1 | 0
Dividing row 2 with 2 and multiplying row 1 with -4 and add it into row 3:
• 1 1 1 | -6
• 0 1 0 | -3
• 0 -2 -3 | 24
Multiplying row 2 with 2 and add it into row 3:
• 1 1 1 | -6
• 0 1 0 | -3
• 0 0 -3 | 18
It can be seen that when this updated augmented matrix is converted into a system, it comes out to be:
• a + b + c = -6
• b = -3
• -3c = 18 (This implies that c = -6.)
Put c = -6 and b = -3 in equation 1:
• a + (-3) + (-6) = -6
• a = -6 + 3 + 6
• a = 3.
So f(x) = (All conditions are being satisfied)!!!
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Answer:
1) Function; Linear Function (y=x)
2) Function; Exponential Function (y=bˣ)
3) Not a function.
Step-by-step explanation:
So, let's go through each table of values. I'm also going to graph them (see attached graphs) because it will make things simpler and provide a visual.
First, also recall what the definition of a function is. A function is a function if and only if an x only has <em>one</em> distinct y-value. No x can have two or more y-values. For instance, (1,2) and (1,3) is <em>not</em> a function between the x repeats. So, let's go through the choices:
1)
We have the table as shown. Let's plot the points. See TABLE 1:
First, there are no repeating x-values, so we can be sure that this is indeed a function.
Looking at the graph, we can see that this resembles a linear graph. So, the parent function of this is a line.
The parent function of a line is:
So, table 1 is associated with a linear function.
2)
Let's again plot the points. See TABLE 2.
Again, there are no repeating x-values, so this is also a function.
Looking at the graph, this resembles an exponential function. The function increases slowly at first but starts increasing increasingly fast. The parent function is thus an exponential function.
So, table 2 is associated with an exponential function.
3)
First, notice how all the x-coordinates are repeating. And since they do <em>not</em> equal the same thing, this is not a function.
Table 3 is not a function and so has no corresponding family function.
