To get Daredevil Danny through the Flaming Hoop Jump of Awesome, you will need to know the values
of a, b, and c, which are the coefficients of the quadratic equation in standard form.
a) What is the equation for a parabola in standard form? (5 points)
b) Describe how to convert from vertex form to standard form. (10 points
The Correct Answer and Explanation is:
a) What is the equation for a parabola in standard form?
The standard form of a quadratic equation is expressed as:
[
y = ax^2 + bx + c
]
In this equation:
- (y) is the output variable (dependent variable).
- (x) is the input variable (independent variable).
- (a), (b), and (c) are coefficients, where:
- (a) determines the direction (upward if (a > 0) and downward if (a < 0)) and the width of the parabola.
- (b) affects the position of the vertex along the x-axis.
- (c) is the y-intercept of the parabola (the point where the graph intersects the y-axis).
b) Describe how to convert from vertex form to standard form.
The vertex form of a quadratic equation is given by:
[
y = a(x – h)^2 + k
]
In this form, ((h, k)) represents the vertex of the parabola. To convert from vertex form to standard form, you need to expand the equation and simplify it. Here are the steps involved:
- Start with the vertex form equation: [
y = a(x – h)^2 + k
] - Expand the squared term: Use the identity ((x – h)^2 = x^2 – 2hx + h^2). Substituting this into the equation gives: [
y = a(x^2 – 2hx + h^2) + k
] - Distribute the coefficient (a): Multiply each term inside the parentheses by (a): [
y = ax^2 – 2ahx + ah^2 + k
] - Combine the constant terms: The constant term in the standard form (c) can be represented as (c = ah^2 + k). Therefore, the equation now looks like: [
y = ax^2 – 2ahx + (ah^2 + k)
] - Write the final standard form: The equation in standard form is: [
y = ax^2 + bx + c
] where (b = -2ah) and (c = ah^2 + k).
Explanation of the Conversion Process
Converting from vertex form to standard form allows you to analyze the quadratic equation more effectively. The vertex form highlights the vertex of the parabola, which is essential for graphing and understanding the parabola’s shape and position. On the other hand, the standard form makes it easier to identify the coefficients that define the parabola’s properties, such as its width and direction.
The process involves algebraic manipulation, specifically expanding and rearranging terms. The first step is to expand the squared term, which involves using the identity for squaring a binomial. Distributing the (a) coefficient is crucial, as it affects all terms within the parentheses. Finally, combining constant terms ensures that you have a single representation for (c), which is critical when determining the y-intercept of the parabola. Understanding these forms and the conversion process helps in solving problems related to quadratic functions, analyzing their graphs, and finding their roots or zeros.