Problem 87 - Algebra 2

Question

Use the following information about quadratic functions for Exercises \(85-90\) . vertex form: \(y=a(x-h)^{2}+k \quad\) standard form: \(y=a x^{2}+b x+c\) When \(y=-2(x+3)^{2}+25\) is written in standard form, what is the value of \(c ?\)

Step-by-Step Solution

Answer

The value of \(c\) when the equation is written in standard form is \(7\).

Step 1: Identify the Vertex Form of the Quadratic Equation

Recognize that the given equation is in the vertex form of a quadratic, which is given by the template equation: \(y = a(x - h)^2 + k\). The given equation is: \(y=-2(x + 3)^2 + 25\).

Step 2: Expand the Vertex Form into the Standard Form

Expand the square in the vertex form equation to convert it into the standard form. Start by applying the square to the binomial: \((-2(x + 3)^2 = -2(x^2 + 6x + 9))\).

Step 3: Multiply Through by the Coefficient

Distribute the factor of \(-2\) across all terms in the parenthesis: \(-2(x^2 + 6x + 9) = -2x^2 - 12x - 18\).

Step 4: Add the Vertex y-value

Combine the constant from the expanded form with the \(k\)-value from the vertex form: \(-2x^2 - 12x - 18 + 25\).

Step 5: Combine Like Terms

Combine any like terms to simplify the equation further. In this case, combine the constant terms: \(-18 + 25 = 7\). The standard form of the equation is now \(y = -2x^2 - 12x + 7\).

Step 6: Identify the Value of c

From the standard form of the quadratic equation \(y = ax^2 + bx + c\), identify the value of \(c\). Here, \(c = 7\).

Key Concepts

Vertex Form of a Quadratic

The vertex form of a quadratic function is particularly useful when we need to identify key attributes of a parabola, like its vertex, quickly and easily. With the general expression being y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola, a is a coefficient that affects the width and direction of the parabola, and x is the independent variable.

To proceed from vertex form to standard form, you expand the squared binomial and simplify. Vertex form provides clarity on the parabola's peak or trough, making it straightforward to sketch the graph without extensive calculations.

In educational practices, it's advantageous to encourage students to visualize the parabola by plotting the vertex and assessing the 'a' value's effect on the parabola's shape to solidify understanding.

Standard Form of a Quadratic

Standard form offers a different perspective on quadratics. It's the function expressed as y = ax^2 + bx + c. This form is widely used for solving quadratic equations using methods like factoring, completing the square, or applying the quadratic formula.

The coefficients a, b, and c play distinct roles in determining the parabola's properties: a influences the direction and width of the parabola, b affects the axis of symmetry and vertex horizontal position, and c represents the y-intercept.

Converting between standard form and vertex form enhances students' algebraic manipulation skills and aids in understanding how different forms highlight various aspects of the graph.

Expanding Binomials

Expanding binomials is an essential skill when working with quadratic functions, especially when converting from vertex form to standard form. This process involves applying the perfect square or FOIL (First, Outside, Inside, Last) method.

Consider the binomial (x + a)^2. Its expansion is x^2 + 2ax + a^2. The coefficients in this expansion help to rewrite the quadratic equation in a different form. It's beneficial to emphasize the importance of careful expansion to avoid common errors such as missing the middle term with exercises that require students to expand and simplify binomials step by step.

Quadratic Coefficients

Quadratic coefficients a, b, and c are much more than just numbers; they dictate the shape and position of a parabola in a quadratic function. The a coefficient controls the parabola's concavity and width, b influences the symmetry around the vertex, and c gives the y-intercept.

Recognizing the significance of each coefficient enhances students' comprehension of quadratic functions' behavior. For instance, highlighting that a positive 'a' results in a 'smiling' parabola, while a negative 'a' gives us a 'frowning' parabola, can help students remember how the value of 'a' affects the graph. When 'c' is isolated as the y-intercept, it can ground students' understanding of the graph's intersection with the y-axis.