The vertex of the quadratic is (2 , 12)
Step-by-step explanation:
The vertex form of the quadratic function is f(x) = a(x - h)² + k, where
∵ The function of quadratic
∴ f(x) = a(x - h)² + k
∵ The axis of symmetry of the function is x = 2
∵ The axis of symmetry of a quadratic function is a vertical line at x = h
∴ h = 2
- Substitute h by 2 in f(x)
∴ f(x) = a(x - 2)² + k
∵ The quadratic passing through the point f(3/2) = 9
- That means the coordinates of the point are [tex](\frac{3}{2},9)[/tex]
∴ x = [tex]\frac{3}{2}[/tex] and y = 9
- Substitute x and y in the equation by the values above
∴ 9 = a( [tex]\frac{3}{2}[/tex] - 2)² + k
∴ 9 = a( [tex]\frac{-1}{2}[/tex] )² + k
∴ 9 = [tex]\frac{1}{4}[/tex] a + k
- Multiply each term by 4
∴ 36 = a + 4 k
∴ a + 4 k = 36 ⇒ (1)
∵ x intercept of f(x) = 3
- That means the graph of f(x) intersects x-axis at point (3 , 0)
∴ x = 3 and y = 0
- Substitute x and y in the equation by the values above
∴ 0 = a(3 - 2)² + k
∴ 0 = a(1)² + k
∴ 0 = a + k
∴ a + k = 0 ⇒ (2)
Now we have system of equation let us solve it to find k
∵ a + 4 k = 36 ⇒ (1)
∵ a + k = 0 ⇒ (2)
- Subtract equation (2) from equation (1) to eliminate a
∴ 3 k = 36
- Divide both sides by 3
∴ k = 12
∵ (h , k) are the coordinates of the vertex of the quadratic function
∴ (2 , 12) are the coordinates of the vertex of the quadratic function
The vertex of the quadratic is (2 , 12)
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