Sorry, I forgot to enter all the info before. Below is the updated info for the problem.
What's the best way to solve functions? I can do basic ones, but have trouble with the harder ones.
I'm having trouble with the following:
Inputs x 1, 2, 3, 4, 5
Outputs y 0, 2, 6, 12, 20
I have tried everything and can't come up with the equation. Please help! Also, if you have any tips on how to solve that would be great. Thanks!
To solve a function, we need to find the equation that relates the inputs (x values) to the outputs (y values). In this case, you have a set of input-output pairs:
Input (x): 1, 2, 3, 4, 5
Output (y): 0, 2, 6, 12, 20
To find the equation, we can look for patterns or relationships between the inputs and outputs. Let's examine the differences between consecutive output values:
Difference (Δy): 2, 4, 6, 8
We notice that the differences between consecutive output values are increasing by 2 each time. This indicates that there might be a quadratic relationship between the inputs and outputs.
To confirm this, let's examine the differences between consecutive differences:
Difference of Differences (Δ²y): 2, 2, 2
The differences of differences are constant, indicating a quadratic relationship. Moreover, the fact that the second-degree differences are constant suggests that the equation is quadratic.
Now, we can start building the equation step by step.
Step 1: Start with a general quadratic equation: y = ax² + bx + c
Step 2: Substitute the given input-output pairs into the equation. For example, when x = 1, y = 0:
0 = a(1²) + b(1) + c
Step 3: Repeat step 2 with other input-output pairs to create a system of equations. Using the given values, we get:
2 = a(2²) + b(2) + c
6 = a(3²) + b(3) + c
12 = a(4²) + b(4) + c
20 = a(5²) + b(5) + c
Step 4: Solve the system of equations simultaneously to find the values of a, b, and c.
There are several methods to solve systems of equations, such as substitution, elimination, or matrices.
One approach is to substitute the value of c from the first equation into the remaining equations. This will eliminate the variable c from the system, and then you can solve for a and b.
After solving the system, you will obtain the values for a, b, and c which you can use to write the final equation in the form y = ax² + bx + c.