Question

write an equation for the parabola in standard form.

1. - / 1 points

write an equation for the parabola in standard form.

1. - / 1 points

write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point. (let x be the independent the dependent variable.)
vertex: (2, -1); point: (-1, 17)

Explanation:

Problem 12 (Parabola with vertex \((6, 0)\) and point \((7, 2)\))

Step1: Recall standard form of parabola

The standard form of a parabola with vertex \((h, k)\) is \(y = a(x - h)^2 + k\). Here, vertex is \((6, 0)\), so \(h = 6\), \(k = 0\). The equation becomes \(y = a(x - 6)^2 + 0\) or \(y = a(x - 6)^2\).

Step2: Substitute the point \((7, 2)\)

Substitute \(x = 7\), \(y = 2\) into the equation: \(2 = a(7 - 6)^2\). Simplify: \(2 = a(1)^2\) so \(a = 2\).

Step3: Write the final equation

Substitute \(a = 2\) back into \(y = a(x - 6)^2\): \(y = 2(x - 6)^2\). Expand if needed: \(y = 2(x^2 - 12x + 36) = 2x^2 - 24x + 72\). But standard form is \(y = a(x - h)^2 + k\), so \(y = 2(x - 6)^2\) (or expanded form \(y = 2x^2 - 24x + 72\)).

Step1: Recall standard form

Standard form: \(y = a(x - h)^2 + k\), vertex \((h, k) = (2, -1)\), so equation is \(y = a(x - 2)^2 - 1\).

Step2: Substitute the point \((-1, 17)\)

Substitute \(x = -1\), \(y = 17\): \(17 = a(-1 - 2)^2 - 1\). Simplify: \(17 = a(-3)^2 - 1\) → \(17 = 9a - 1\). Add 1: \(18 = 9a\) → \(a = 2\).

Step3: Write the final equation

Substitute \(a = 2\) into \(y = a(x - 2)^2 - 1\): \(y = 2(x - 2)^2 - 1\). Expand: \(y = 2(x^2 - 4x + 4) - 1 = 2x^2 - 8x + 8 - 1 = 2x^2 - 8x + 7\). Standard form is \(y = 2(x - 2)^2 - 1\) (or expanded \(y = 2x^2 - 8x + 7\)).

Step1: Standard form with vertex \((-2, -5)\)

Standard form: \(y = a(x - h)^2 + k\), \(h = -2\), \(k = -5\), so \(y = a(x + 2)^2 - 5\).

Step2: Substitute \((0, -1)\)

Substitute \(x = 0\), \(y = -1\): \(-1 = a(0 + 2)^2 - 5\). Simplify: \(-1 = 4a - 5\). Add 5: \(4 = 4a\) → \(a = 1\).

Step3: Final equation

Substitute \(a = 1\): \(y = (x + 2)^2 - 5\). Expand: \(y = x^2 + 4x + 4 - 5 = x^2 + 4x - 1\). Standard form is \(y = (x + 2)^2 - 5\) (or \(y = x^2 + 4x - 1\)).

Answer:

\(y = 2(x - 6)^2\) (or \(y = 2x^2 - 24x + 72\))

Problem 13 (Vertex \((2, -1)\) and point \((-1, 17)\))