Problem 59 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c\). Use your calculator to support your results. Vertex \((-4,-2) ;\) through \((2,-26)\)

Step-by-Step Solution

Answer

The equation is \(P(x) = -\frac{2}{3}x^2 - \frac{16}{3}x - \frac{38}{3}\).

Step 1: Understanding the Vertex Form

To find the equation of the quadratic function, start by recognizing that a quadratic equation can be expressed in vertex form as \(P(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola.

Step 2: Substitute the Vertex

Since the given vertex is \((-4, -2)\), substitute \(h = -4\) and \(k = -2\) in the vertex form. The equation becomes \(P(x) = a(x+4)^2 - 2\).

Step 3: Use the Point on the Parabola

The quadratic passes through the point \((2, -26)\). Substitute \(x = 2\) and \(P(x) = -26\) into the equation \(-26 = a(2+4)^2 - 2\).

Step 4: Solve for \(a\)

Solve the equation from the previous step: \(-26 = a(6)^2 - 2\). This becomes \(-26 = 36a - 2\). Add 2 to both sides: \(-24 = 36a\), then divide by 36 to find \(a = -\frac{2}{3}\).

Step 5: Write the Quadratic in Standard Form

Substitute \(a = -\frac{2}{3}\) back into the vertex form equation: \(P(x) = -\frac{2}{3}(x+4)^2 - 2\). Expand \(P(x) = -\frac{2}{3}(x^2 + 8x + 16) - 2\) to get \(P(x) = -\frac{2}{3}x^2 - \frac{16}{3}x - \frac{32}{3} - 2\). Combine constants: \(-\frac{32}{3} - 2 = -\frac{32}{3} - \frac{6}{3} = -\frac{38}{3}\).

Step 6: Express in Standard Form

The quadratic equation in standard form is \(P(x) = -\frac{2}{3}x^2 - \frac{16}{3}x - \frac{38}{3}\).

Key Concepts

Vertex Form

The vertex form of a quadratic function is a great way to understand the geometry of the parabola. It's written as \( P(x) = a(x-h)^2 + k \). In this form, \((h, k)\) represents the vertex of the parabola.
The parameter \(a\) affects the width and direction of the parabola's opening. If \(a > 0\), the parabola opens upwards, if \(a < 0\), it opens downwards.

• Vertex Position: The position of the vertex is directly indicated by \(h\) and \(k\) in the equation. This makes it very convenient when the vertex is known, as in our original exercise.
• Transformations: Changing \(h\) shifts the parabola left or right, while changing \(k\) shifts it up or down.

This form is particularly useful when you want to quickly identify the vertex and understand how the parabola moves in the coordinate plane.

Standard Form

The standard form of a quadratic function is expressed as \( P(x) = ax^2 + bx + c \). This form is often used in algebra to perform operations like adding, subtracting, and solving quadratic equations. It is also the end goal in many exercises, as seen in our original solution.

• Coefficients: Here, \(a\) defines the parabola's opening direction and width, just like in vertex form. The coefficients \(b\) and \(c\) help in determining the specific curve of the parabola on the graph.
• Roots: The standard form makes it easier to use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), to find the roots of the equation.

When converting from vertex form to standard form, we expand the squared term to derive an equivalent polynomial, as shown in the step-by-step solution.

Parabola

A parabola is a symmetrical curve that one usually encounters in quadratic functions. It represents the graph of a quadratic equation and is a central object of study in algebra and calculus. The shape of a parabola is defined by its focus and directrix, but for most algebra problems, understanding its basic properties suffices.

• Symmetry: Parabolas have an axis of symmetry, a vertical line passing through the vertex. This symmetry makes them predictable and useful for modeling.
• Direction: As mentioned, the sign of \(a\) in both the vertex and standard form determines if it opens up or down.
• Vertex: The vertex is the highest or lowest point depending on the parabola's orientation, essential for determining maxima or minima in optimization problems.

Understanding parabolas is critical when working with projectiles in physics, optimizing areas, and interpreting graphs in economics.