Question

1. the vertex of a parabola is (1.5, 9) and the zeros are (1, 0) and (2, 0). graph the parabola and write an equation of a parabola that goes through these three points.

Explanation:

Step1: Choose vertex form of parabola

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Given vertex \((1.5, 9)\), so \( h = 1.5 \), \( k = 9 \). The equation becomes \( y = a(x - 1.5)^2 + 9 \).

Step2: Substitute a zero into the equation

We know a zero is \((1, 0)\), substitute \( x = 1 \), \( y = 0 \) into the equation: \( 0 = a(1 - 1.5)^2 + 9 \).

Step3: Solve for \( a \)

Simplify the equation: \( 0 = a(-0.5)^2 + 9 \) → \( 0 = 0.25a + 9 \). Subtract 9 from both sides: \( -9 = 0.25a \). Divide by 0.25: \( a = -36 \).

Step4: Write the final equation

Substitute \( a = -36 \) back into the vertex form: \( y = -36(x - 1.5)^2 + 9 \). We can also expand it: \( y = -36(x^2 - 3x + 2.25) + 9 = -36x^2 + 108x - 81 + 9 = -36x^2 + 108x - 72 \). For graphing, plot the vertex \((1.5, 9)\) and the zeros \((1, 0)\), \((2, 0)\), then sketch the parabola opening downward (since \( a < 0 \)).

Answer:

The equation of the parabola is \( y = -36(x - 1.5)^2 + 9 \) (or \( y = -36x^2 + 108x - 72 \)) and the graph has vertex \((1.5, 9)\) and passes through \((1, 0)\) and \((2, 0)\) opening downward.