Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

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Step-by-Step Solution

Answer

The equation is \( y = -\frac{1}{3}(x + 3)^2 + 2 \).

Step 1: Understand the Problem

The problem asks us to determine the quadratic function that has vertex (-3, 2) and passes through the point (0, -1). A quadratic function can be expressed in the vertex form: \( y = a(x-h)^2 + k \), where (h, k) is the vertex. Here, the vertex is (-3, 2).

Step 2: Write the Vertex Form Equation

Using the vertex provided, we have \( h = -3 \) and \( k = 2 \) in the vertex form equation. So, our equation becomes \( y = a(x + 3)^2 + 2 \).

Step 3: Use the Point to Find 'a'

We use the point (0, -1) to find the value of \( a \). Substitute \( x = 0 \) and \( y = -1 \) into the equation: \( -1 = a(0 + 3)^2 + 2 \).

Step 4: Solve for 'a'

Simplify the equation: \( -1 = 9a + 2 \). Subtract 2 from both sides to get \( -3 = 9a \). Divide both sides by 9 to solve for \( a \): \( a = -\frac{1}{3} \).

Step 5: Write the Final Quadratic Equation

Substitute \( a = -\frac{1}{3} \) back into the vertex form equation to get the final equation of \( y = -\frac{1}{3}(x + 3)^2 + 2 \).

Key Concepts

Quadratic Function

A quadratic function is one of the fundamental concepts in algebra. It is a polynomial function of degree 2, which means its highest power of the variable is 2. The standard form of a quadratic function is expressed as:

• \( y = ax^2 + bx + c \)

where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The graph of a quadratic function is a curve called a parabola, which can open either upwards or downwards depending on the sign of the coefficient \( a \).
A key feature of this function is its vertex, which serves as the main focus in its graph's symmetry.

Vertex

The vertex of a quadratic function is a crucial point, as it determines the peak or the lowest point of the parabola. In terms of the vertex form of a quadratic equation:

• \( y = a(x-h)^2 + k \)

It is represented by the point \((h, k)\). This point acts as the "turning point" of the parabola.

• If \( a > 0 \), the parabola opens upwards, making the vertex the minimum point.
• If \( a < 0 \), it opens downwards, making it the maximum point.

Understanding the vertex allows you to graph the parabola accurately and identify important properties like axis of symmetry, which is the vertical line that passes through the vertex, given by \( x = h \).

Quadratic Equation

A quadratic equation is any equation that can be rearranged into the form of a quadratic function. These equations frequently appear in various algebraic contexts and have a multitude of solving methods.

• Standard form: \( ax^2 + bx + c = 0 \)
• Vertex form: \( y = a(x-h)^2 + k \)
• Factored form: \( y = a(x-p)(x-q) \)

To convert a quadratic equation from one form to another, algebraic techniques such as completing the square or factoring are used. These equations can have two, one, or no real solutions, which can be found using the quadratic formula or factoring, depending on the discriminant \( (b^2 - 4ac) \).

Coordinates

Coordinates are vital in pinpointing the exact location of points on a graph. In the context of quadratic functions, they help define the vertex and any other point that the parabola passes through. The coordinate system operates on an \( x \) (horizontal) and \( y \) (vertical) axis.

• Vertically, the vertex coordinates are \((h, k)\), which also tells us whether the parabola some level above or below the x-axis.
• Coordinates are used to determine specific points, such as where the parabola crosses the y-axis or any point used to solve for unknowns like \( a \) in vertex form.

Using these coordinate points in the quadratic equation lets you find unknown parameters and characteristics of the graph.

Algebra

Algebra is the branch of mathematics concerning symbols and the rules for manipulating these symbols. It's the foundational system behind all quadratic functions and equations. In quadratic problems, algebra helps us:

• Write equations in canonical forms like vertex form, which is especially useful for graph plotting.
• Use operations such as addition, subtraction, and factorization to solve equations.
• Substitute known values to find unknown quantities, like solving for \( a \) given a vertex and a point.

Algebraic processes, such as expanding, factoring, or completing the square, aid in transforming the equation to make calculations feasible, highlighting how deeply integrated algebra is in solving and understanding quadratic relationships.