Answer: f(x) = (x - 3)²
<u>Step-by-step explanation:</u>
Use the vertex formula: y = a(x - h)² + k
and input the vertex (h, k) = (3, 0) and the given point (x, y) = (4, 2) to solve for a
2 = a(4 - 3)² + 0
2 = a(1)²
2 = a
Now, input (h, k) = (3, 0) and a = 1 into the vertex formula:
y = 1(x - 3)² + 0 → y = (x - 3)²
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Answer: g(x) = -(x + 1.25)² + 2.5625
<u>Step-by-step explanation:</u>
Use the vertex formula: y = a(x - h)² + k
and input the given points for (x, y) to create a system of equations, then solve for a, h, and k.
EQ1: 2 = a(-2 - h)² + k
2 = a(4 + 4h + h²) + k
2 - 4a - 4ah - ah² = k
EQ2: 1 = a(0 - h)² + k
1 = ah² + k
1 - ah² = k
EQ3: -2.5 = a(1 - h)² + k
-2.5 = a(1 - 2h + h²) + k
-2.5 -a + 2h + ah² = k
<u>Substitute</u> - set EQ1 = EQ2 and EQ2 = EQ3 to eliminate k
EQ1 = EQ2: 2 - 4a - 4ah - ah² = 1 - ah²
1 - 4a - 4ah = 0
EQ2 = EQ3: 1 - ah² = -2.5 - a + 2ah - ah²
3.5 + a - 2ah = 0
<u>Elimination:</u> - now solve the system for "a"
1 - 4a - 4ah = 0 → 1(1 - 4a - 4ah = 0) → 1 - 4a - 4ah = 0
3.5 + a - 2ah = 0 → -2(3.5 + a - 2ah = 0) → <u> -7 - 2a + 4ah = 0 </u>
-6 - 6a = 0
-6a = 6
<h2>
a = -1</h2>
Next, replace "a" with -1 into either of the equations to solve for "h"
1 - 4a - 4ah = 0
1 - 4(-1) - 4(-1)h = 0
1 + 4 + 4h = 0
5 + 4h = 0
4h = -5
<h2> h = -1.25</h2>
Now, replace "a" with -1 and "h" with -1.25 into any of the original equations (EQ1, EQ2, or EQ3) to solve for k:
1 - ah² = k
1 - (-1)(-1.25)² = k
1 - (-1)(1.5625) = k
1 + 1.5625 = k
<h2> 2.5625 = k</h2>
Now, input (h, k) = (-1.25, 2.5625) and a = -1 into the vertex formula:
y = -1(x - (-1.25))² + 2.5625 → y = -(x + 1.25)² + 2.5625
