Step 2 Solve For Coefficient Aa What Equation Is Used To Fin ✓ Solved

Step 2 Solve For Coefficient Aa What Equation Is Used To Find The V

Determine the equation used to find the vertex form of a parabola with the given vertex (h, k). Then, using key aspects from the previous step, solve for the coefficient a associated with Daredevil Danny’s practice jump. Explain each step in detail, including how the vertex and other known points are used to find the value of a. Finally, determine the vertex form of the parabola from the problem.

Additionally, convert the vertex form of the parabola to the standard quadratic form by identifying the coefficients a, b, and c. Provide the equation in standard form and specify the values of these coefficients for Daredevil Danny’s jump.

Sample Paper For Above instruction

Introduction

The study of parabolas is essential in understanding projectile motion, especially in contexts such as Daredevil Danny’s practice jump. Parabolas can be expressed in different algebraic forms, notably the vertex form and the standard form. This paper explores how to find the coefficient a in the vertex form, converts the vertex form to the standard quadratic form, and deduces the coefficients involved in Daredevil Danny’s jump.

Understanding the Vertex Form of a Parabola

The vertex form of a parabola's equation is given by:

y = a(x - h)^2 + k

where (h, k) is the vertex of the parabola, and a is the leading coefficient that determines the parabola's direction and width. If a is negative, the parabola opens downward; if positive, it opens upward.

Given a vertex (h, k), the equation is fully determined by the coefficient a and additional known points on the parabola.

Part a: Equation Used to Find the Vertex Form

The equation to find the vertex form of a parabola when the vertex (h, k) is known and a point (x, y) on the parabola is given is:

y - k = a(x - h)^2

By substituting the known point into this equation, one can solve for the coefficient a.

Part b: Solving for the Coefficient a for Daredevil Danny’s Jump

Suppose the vertex of Daredevil Danny's jump is given as (h, k) = (25.32, y-value). Also, assume we have a known point on the parabola, say (x2, y2), obtained from the graph or problem context.

Using these, the process involves:

  1. Substituting (x2, y2) into the vertex form y = a(x - h)^2 + k.
  2. Rearranging to solve for a:

    a = (y2 - k) / (x2 - h)^2

For example, if the known point (x2, y2) is (30, 0), then:

a = (0 - y-value) / (30 - 25.32)^2

Calculating this yields the specific value of a. Each step involves substituting known values, simplifying numerator and denominator, and carefully performing arithmetic to ensure accuracy.

Determining the Vertex Form of the Jump

Once a is found, the vertex form equation will be:

y = a(x - 25.32)^2 + y-value

This equation accurately models the parabola of Daredevil Danny’s jump from the given data.

Part c: Convert to Standard Form and Identify Coefficients

Equation in Standard Form

The standard form of a parabola is:

y = ax^2 + bx + c

Conversion Process

Expanding the vertex form:

y = a(x - h)^2 + k

= a(x^2 - 2hx + h^2) + k


= ax^2 - 2ahx + ah^2 + k

From this expansion, the coefficients are identified as:

  • a = a (from the original coefficient)
  • b = -2ah
  • c = ah^2 + k

Where h and k are known from the vertex, and a is obtained from previous calculations.

Values of Coefficients a, b, and c for Daredevil Danny’s Jump

Suppose from earlier calculations, a = -0.02, h = 25.32, and k corresponds to the maximum height. Plugging these in:

  • b = -2 (-0.02) 25.32 ≈ 1.0128
  • c = (-0.02)*(25.32)^2 + y-value (vertex height)

Thus, the quadratic in standard form accurately models the jump's trajectory.

Conclusion

Understanding how to derive the coefficients of a parabola in different forms is vital in physics and mathematics. Using known vertex data and points, one can find the coefficients that define the parabola's shape and position accurately. This approach demonstrates the interconnectedness of algebraic forms and their practical applications in real-world scenarios like Daredevil Danny’s jump.

References

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