Problem 26 - College Algebra with Corequisite Support

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. $$ (h, k)=(2,0),(x, y)=(4,4) $$

Step-by-Step Solution

Answer

The general form of the quadratic function is \(f(x) = x^2 - 4x + 4\).

Step 1: Understand the Vertex Form of a Quadratic Equation

A quadratic function can be written in the vertex form as \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola. In this problem, the vertex is given as \((2,0)\), so the equation becomes \(f(x) = a(x-2)^2 + 0\), which simplifies to \(f(x) = a(x-2)^2\).

Step 2: Substitute the Point into the Vertex Form

We are given a point on the graph \((x, y) = (4, 4)\). Substitute \(x = 4\) and \(y = 4\) into the vertex form equation \(f(x) = a(x-2)^2\). This results in the equation \(4 = a(4-2)^2\).

Step 3: Solve for the Coefficient 'a'

In the equation from Step 2, \(4 = a(2)^2\). Simplify this to \(4 = a \cdot 4\). Solving for \(a\), we get \(a = 1\).

Step 4: Write the General Form of the Quadratic Equation

Since we have found \(a = 1\), substitute back into the vertex form equation \(f(x) = a(x-2)^2\), resulting in \(f(x) = (x-2)^2\). Expand this equation to convert it to general form: \((x-2)^2 = x^2 - 4x + 4\). The general form of the quadratic function is \(f(x) = x^2 - 4x + 4\).

Key Concepts

Vertex Form

The vertex form of a quadratic equation is a way to express a quadratic function. It is particularly useful because it clearly shows both the vertex of the parabola and the orientation of the graph. The vertex form is given by the formula \( f(x) = a(x-h)^2 + k \), where:

• \((h, k)\) is the vertex, indicating the highest or lowest point on the graph depending on the orientation.
• The parameter \(a\) affects the width and the direction of the parabola's opening.

In the example provided, the vertex \((h, k)\) is \((2, 0)\), which means that you can directly write the function in the vertex form as \( f(x) = a(x-2)^2 + 0 \). Simplified further, this becomes \( f(x) = a(x-2)^2 \). This form is particularly advantageous for quickly identifying the vertex on a graph, making it very user-friendly for sketching quadratic functions.

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree. It is most commonly arranged in standard form as \( ax^2 + bx + c = 0 \). Quadratic equations are essential in various fields of science and engineering because they model a wide range of phenomena. Quadratic functions graph into a curve called a parabola.In the exercise, the quadratic function initially starts in the vertex form \( f(x) = a(x-2)^2 \). By expanding this, we reach the general form \( f(x) = x^2 - 4x + 4 \). Here, the coefficients \( a = 1 \), \( b = -4 \), and \( c = 4 \) are easily identifiable once the equation is expanded. This general form is often preferred for straightforward computation and solving purposes, particularly in algebraic manipulations such as factoring or using the quadratic formula.

Parabola

A parabola is the U-shaped curve you get when you graph a quadratic function. It is symmetrical and contains a vertex. Characteristics to note:

• The parabola opens upwards if \(a > 0\) and downwards if \(a < 0\).
• The vertex is the point \((h, k)\), which serves as the turning point of the parabola.
• The axis of symmetry is the vertical line that passes through the vertex, defined by the equation \(x = h\).

In our scenario, the graph's vertex \((2, 0)\) means the parabola's lowest point (or highest if it opened downward) is at \((2, 0)\). The shape tells us the graph is symmetric over the line \(x = 2\). Grasping these concepts helps in understanding how changes in the components of the quadratic equation affect the graph's shape and position.

Coefficient

Coefficients are the numbers situated in front of the variables in an equation. In the context of a quadratic equation:

• The coefficient \(a\) affects the direction and the width of the parabola.
• The coefficient \(b\) influences the position of the vertex along the x-axis in the general form.
• The coefficient \(c\) can be seen as the y-intercept when the quadratic equation is written in standard form \( ax^2 + bx + c \).

In our given exercise, finding the value of \(a\) is crucial. We used the point \((4, 4)\) on the parabola to determine \(a = 1\) by substituting into the vertex form. Thus, knowing the coefficients helps give a complete picture of how the quadratic function behaves and how it can be graphically represented.