Funcions f(x) and g(x) are defined below.
2 answers:
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Answer:
Step-by-step explanation:
We want to find a function of the form:
We know that it passes through the three points: (-11, 2); (-9, -2); and (-5,14).
In other words, if we substitute -11 for <em>x</em>, we should get 2 for <em>y</em>. So:
Simplify:
We will do it to the two other points as well. So, for (-9, -2):
And similarly:
So, to find our function, we will need to determine the values of a, b, and c.
We essentially have a triple system of equations:
To approach this, we can first whittle it down only using two variables.
So, let's use the First and Second Equation. Let's remove the variable <em>b</em>. To do so, we can use elimination. We can multiply the First Equation by 9 and the Second Equation by -11. This will yield:
Distribute:
Now, we can add the two equations together:
Simplify:
Now, let's do the same using the First and Third Equations. We want to cancel the variable <em>b</em>. So, let's multiply the First Equation by 5 and the Third Equation by -11. So:
Simplify:
Now, let's add the two equations together:
Simplify:
Therefore, we now have the two equations:
Let's cancel the <em>c</em>. So, multiply the First Equation by -3. We don't have to do anything special to the second. So:
Multiply:
Now, add it to the Second Equation:
Add:
Divide both sides by -264. So, the value of <em>a</em> is:
Now, we can use either of the two equations above to obtain <em>c</em>. Let's use the first one. So:
Substitute 1 for <em>a: </em>
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Solve for <em>c</em>. Subtract 198 from both sides and divide by -2:
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So, the value of <em>c</em> is 79.
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Finally, we can find b. We can use any of the three original equations. Let's use the First Equation. So:
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Substitute 1 for <em>a</em> and 79 for <em>c</em> and determine the value of <em>b: </em>
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Therefore, the value of <em>b</em> is 18.
So, our equation is:
<em> </em><em />
Substitute in the values:
Simplify:
Now, let's put this into vertex form. To do so, we will need to complete the square. First, let's group the first two terms together:
To complete the square, we will divide the <em>b</em> term by 2 and then square it.
<em>b</em> is 18. 18/2 is 9 and 9² is 81. Therefore, we will add 81 into our parentheses:
Since we added 81, we must also subtract 81. So:
Subtract:
The grouped terms are a perfect square trinomial. Factor:
And this is in vertex form.
And we are done!