Problem 31 - College Algebra

Question

For the following exercises, use the vertex \((h, k)\) and a point on the graph \((x, y)\) to find the general form of the equation of the quadratic function. $$ (h, k)=(3,2),(x, y)=(10,1) $$

Step-by-Step Solution

Answer

The general form is \( f(x) = -\frac{1}{49}x^2 + \frac{6}{49}x + \frac{89}{49} \).

Step 1: Use the Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is given by \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Substitute \(h = 3\) and \(k = 2\) into this equation, yielding \( f(x) = a(x - 3)^2 + 2 \).

Step 2: Substitute the Given Point into the Equation

We have the point \((x, y) = (10, 1)\). Substitute \(x = 10\) and \(y = 1\) into the equation from Step 1: \( 1 = a(10 - 3)^2 + 2 \).

Step 3: Solve for the Unknown Coefficient 'a'

Simplify the equation from Step 2: \( 1 = a(7)^2 + 2 \) becomes \( 1 = 49a + 2 \). Then, solve for \(a\): \( 49a = 1 - 2 \), which simplifies to \( 49a = -1 \). Divide both sides by 49 to obtain \( a = -\frac{1}{49} \).

Step 4: Write the Equation in Vertex Form

Now that we have \(a = -\frac{1}{49}\) and the vertex \((h, k) = (3, 2)\), the vertex form is \( f(x) = -\frac{1}{49}(x - 3)^2 + 2 \).

Step 5: Convert the Vertex Form to General Form

Expand the vertex form \( f(x) = -\frac{1}{49}(x - 3)^2 + 2 \). Start by expanding \( (x - 3)^2 = x^2 - 6x + 9 \). Then, \( f(x) = -\frac{1}{49}(x^2 - 6x + 9) + 2 = -\frac{1}{49}x^2 + \frac{6}{49}x - \frac{9}{49} + 2 \).

Step 6: Simplify the General Form

Combine \(-\frac{9}{49} + 2\). Converting 2 into a fraction with denominator 49 gives \(2 = \frac{98}{49}\). Thus, \(-\frac{9}{49} + \frac{98}{49} = \frac{89}{49}\). So, the general form is \( f(x) = -\frac{1}{49}x^2 + \frac{6}{49}x + \frac{89}{49} \).

Key Concepts

Vertex Form

The vertex form of a quadratic function is particularly useful for identifying the vertex of the parabola. It's written as \( f(x) = a(x - h)^2 + k \). Here, the point \((h, k)\) represents the vertex of the parabola, which where the graph changes direction. In the given exercise, the vertex is \((3, 2)\). By substituting these values, along with a known point on the parabola, we can find the specific quadratic equation for a given function.
To use the vertex form effectively:

• Identify \((h, k)\) from the given vertex.
• Plug \(h\) and \(k\) into the equation \(f(x) = a(x - h)^2 + k\).
• Use additional points to find the coefficient \(a\).

This method allows us to generate specific quadratic functions that perfectly fit specified conditions.

General Form

Converting quadratic equations to general form is crucial for various algebraic manipulations. The general form is \( ax^2 + bx + c \). This format is handy for determining the roots of the equation using methods like factoring, completing the square, or applying the quadratic formula. In the exercise, we converted the vertex form to the general form:
The steps involved are:

• Expand the vertex form equation to get a polynomial.
• Simplify and rearrange the terms to organize them in descending order of power.

The final result is given in the form \( f(x) = -\frac{1}{49}x^2 + \frac{6}{49}x + \frac{89}{49} \) which clearly shows each term as part of the quadratic structure.

Solving Equations

Solving quadratic equations is at the heart of finding unknown values in quadratic functions. When given a specific point on the parabola, we often need to solve for the coefficient \(a\) that modifies the function's width and direction. Let's break it down step-by-step:
One approach:

• Replace \(x\) and \(f(x)\) with the coordinates of the given point in the vertex form.
• Simplify the equation to isolate \(a\).
• Solve for \(a\) through basic algebra, adjusting the equation to solve for changes.

The accurate calculation of \(a\) is essential for forming the precise quadratic equation that follows the graph's path defined by the given vertex and points.

Quadratic Coefficients

Quadratic coefficients \(a\), \(b\), and \(c\) play significant roles in defining the nature of a quadratic equation. These coefficients influence the shape and position of the parabola:

• The coefficient \(a\) determines the direction (up or down) and the width of the parabola.
• \(b\) and \(c\) influence the position relative to the x-axis and y-intercept.

In general form, \( f(x) = ax^2 + bx + c \), each coefficient affects how the graph is plotted. Understanding their function is crucial for effective graphing and solution predictions, as demonstrated in the solution of the exercise where \(a = -\frac{1}{49}\), \(b = \frac{6}{49}\), and \(c = \frac{89}{49}\). These values were calculated to align the function correctly with the known vertex and point.