Math Problem Statement

Sketch the parabola using the given information. Vertex (0,3), Point (-2,-8).

Solution

To graph the parabola using the given information:

Given:

  • Vertex: (0, 3)
  • Point on the parabola: (2,8)(-2, -8)

Step-by-Step Process:

  1. Identify the form of the parabola's equation: Since you are given the vertex, the vertex form of a quadratic equation is: y=a(xh)2+ky = a(x - h)^2 + k where (h,k)(h, k) is the vertex of the parabola.

  2. Substitute the vertex coordinates: Given the vertex is (0, 3), substitute h=0h = 0 and k=3k = 3: y=a(x0)2+3ory=ax2+3y = a(x - 0)^2 + 3 \quad \text{or} \quad y = ax^2 + 3

  3. Use the additional point to find the value of aa: Substitute the coordinates of the point (2,8)(-2, -8) into the equation: 8=a(2)2+3-8 = a(-2)^2 + 3 Simplify the equation to solve for aa: 8=4a+3-8 = 4a + 3 Subtract 3 from both sides: 11=4a-11 = 4a Divide by 4: a=114a = -\frac{11}{4}

  4. Write the final equation: The equation of the parabola is: y=114x2+3y = -\frac{11}{4}x^2 + 3

  5. Graph the parabola:

    • Plot the vertex at (0, 3).
    • Plot the point (2,8)(-2, -8).
    • Since a<0a < 0, the parabola opens downward.
    • Draw a smooth curve through the points.

You can use the graphing tool provided to plot the vertex and the point, then draw the parabola based on these points.

Would you like a more detailed explanation of any of these steps?

Here are some related questions:

  1. What is the standard form of a quadratic equation?
  2. How does the sign of the coefficient aa affect the direction of the parabola?
  3. What are other forms of the quadratic equation besides vertex form?
  4. How do you find the axis of symmetry in a parabola?
  5. Can you determine the roots of the parabola from the vertex form?

Tip: Always verify the direction of your parabola by checking the sign of the leading coefficient aa. If aa is negative, the parabola opens downwards; if positive, it opens upwards.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Graphing Parabolas

Formulas

Vertex form of a quadratic equation: y = a(x - h)^2 + k

Theorems

Vertex form of a parabola

Suitable Grade Level

Grades 9-11