Problem 56 - Introductory Algebra for College Students

Question

Describe how to find a parabola's vertex.

Step-by-Step Solution

Answer

The vertex of a parabola in standard form, \(y = ax^2 + bx + c\), is given by the point \((- \frac{b}{2a}, f(- \frac{b}{2a})\), where \(f(x)\) is the function defined by the standard form equation.

Step 1: Identify 'a' and 'b' values

Read off the coefficients 'a' and 'b' from the standard form of the parabola. The 'a' value is the coefficient of the \(x^2\) term, and 'b' is the coefficient of the 'x' term.

Step 2: Apply Vertex Formula

Insert the values of 'a' and 'b' into the formula for the x-coordinate of the vertex: \(- \frac{b}{2a}\). This will provide the x-coordinate of the vertex.

Step 3: Find the y-coordinate of the Vertex

Replace 'x' in the equation of the parabola (Standard form) with the x-coordinate calculated in Step 2. Simplify this to find the y-coordinate of the vertex.

Key Concepts

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is a way to express quadratic functions in a familiar and consistent way. It is usually written as: \[y = ax^2 + bx + c\] This equation represents a parabola, which is a U-shaped curve that can open upwards or downwards.

• The coefficient 'a' determines the direction of the parabola. If 'a' is positive, the parabola opens upwards. If 'a' is negative, it opens downwards.
• The coefficient 'b' affects the tilt or the direction in which the parabola opens from its vertex.
• The constant 'c' moves the parabola up or down on the y-axis.

Understanding the standard form is crucial because it provides essential information about the parabola's shape, direction, and position in a single glance.

Vertex Formula

The vertex of a parabola is a key point representing the minimum or maximum of the curve, depending on the direction the parabola opens. To find the vertex, especially from a quadratic equation in standard form, the vertex formula is used. The vertex's x-coordinate is calculated using the formula:\[x = -\frac{b}{2a}\]

• This formula comes from completing the square or using calculus techniques.
• It is derived by setting the derivative of the quadratic equation to zero, finding the point where the slope is zero, which is the vertex.

Once you have the x-coordinate, substitute it back into the original quadratic equation to find the y-coordinate. This gives you the exact vertex point \((x, y)\). Finding this point is essential for graphing the parabola and understanding its properties.

Identifying Coefficients in a Quadratic

To utilize the standard form and vertex formula correctly, accurately identifying the coefficients 'a' and 'b' is necessary. In the quadratic equation \[y = ax^2 + bx + c\]

• 'a' is the number directly in front of the \(x^2\) term. It is the leading coefficient and influences both the direction and the stretch of the parabola.
• 'b' is the coefficient of the 'x' term. It affects the slope or the tilting direction of the parabola as it passes through the vertex.

Incorrect identification of these coefficients will lead to errors in calculating the vertex because they are critical in the vertex formula \(x = -\frac{b}{2a}\). Carefully reading and understanding these coefficients from the quadratic expression ensures accurate graphing and deeper comprehension of the parabola's characteristics.