Answer: y = 2(x-5)(x-7)
x intercepts are 5 and 7
The third point I used was (6,-2); however, you can use any other point you want as long as it's not one of the x intercepts mentioned.
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Explanation:
The x intercepts are 5 and 7, where the parabola crosses the x axis. Because of these roots, the two factors are x-5 and x-7.
We can think of the root x = 5 turning into x-5 = 0 after subtracting 5 from both sides. The same goes for x = 7 turning into x-7 = 0.
Put together, the two factors lead to (x-5)(x-7) and we get the equation y = (x-5)(x-7). Plugging either x = 5 or x = 7 into this equation produces y = 0.
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Notice that plugging in x = 6 leads to...
y = (x-5)(x-7)
y = (6-5)(6-7)
y = (1)(-1)
y = -1
However, the graph shows that x = 6 should lead to y = -2 instead. To fix this, we can stick a 2 out front like so
y = 2(x-5)(x-7)
y = 2(6-5)(6-7)
y = 2(1)(-1)
y = -2
The error has been fixed. As you can see, we need this third point to help nail down the parabolic equation. You can pick any other x value you want as long as it's not 5 or 7.
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Let's try another x value. Let's say we pick x = 8
y = 2(x-5)(x-7)
y = 2(8-5)(8-7)
y = 2(3)(1)
y = 6
We see that x = 8 leads to y = 6. This matches what the graph shows that (8,6) is a point on the parabola. I'll let you try other x values.
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In short, we used the roots 5 and 7 to generate the partial factorization (x-5)(x-7). Then we used a third point (6,-2) to help determine the coefficient out front which was 2.
So after using that point, we go from y = (x-5)(x-7) to y = 2(x-5)(x-7) which is the final answer. This is considered x-intercept form because we can quickly read off the x intercepts from this equation.
