Problem 26 - 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. $$ (h, k)=(2,0),(x, y)=(4,4) $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Vertex Form

The vertex form of a quadratic equation is \( f(x) = a(x - h)^2 + k \). Here, \((h, k)\) is the vertex of the parabola.

Step 2: Substitute the Vertex Coordinates

Substitute the given vertex \((h, k) = (2, 0)\) into the vertex form equation. This gives us \( f(x) = a(x - 2)^2 + 0 \), which simplifies to \( f(x) = a(x - 2)^2 \).

Step 3: Substitute the Point into the Equation

Use the point \((x, y) = (4, 4)\) and substitute these values into the equation to find \(a\). We have \(4 = a(4 - 2)^2\).

Step 4: Solve for 'a'

Solve the equation \(4 = a(2)^2\). This simplifies to \(4 = 4a\). Divide both sides by 4 to find \(a = 1\).

Step 5: Write the Equation in Vertex Form

Now that we know \(a = 1\), substitute back into the vertex form: \( f(x) = 1(x - 2)^2\), which simplifies to \( f(x) = (x - 2)^2 \).

Step 6: Convert to General Form

Expand \((x - 2)^2\) to convert the equation to general form: \( f(x) = x^2 - 4x + 4 \).

Key Concepts

Vertex Form

The vertex form of a quadratic equation is an important concept that helps in understanding the shape and positioning of a parabola on the coordinate plane. We express it as \( f(x) = a(x - h)^2 + k \). Here,

• \( (h, k) \) is the vertex of the parabola, which is the highest or lowest point depending on the orientation.
• The parameter \( a \) affects the width and direction of the parabola. If \( a \) is positive, the parabola opens upwards, while a negative \( a \) means it opens downwards.

In our exercise, we were provided with a vertex \((h, k) = (2, 0)\). This means the point \( (2, 0) \) is the tip of our parabola. Using vertex form allows us to easily derive the specific equation of a parabola by plugging in these vertex coordinates.
By substituting the vertex into the vertex form, we simplify our equation to \( f(x) = a(x - 2)^2 \), indicating the parabola's vertex is at \( x = 2 \). This representation is beneficial because it visually highlights the vertex on the graph.

Parabola

A parabola is a U-shaped curve that represents the graph of a quadratic equation. It's symmetric around its axis of symmetry, which vertically passes through the vertex.

• The direction in which a parabola opens depends on the sign of the coefficient \( a \) in the vertex form \( f(x) = a(x - h)^2 + k \).
• If \( a > 0 \), the parabola opens upwards, resembling a smile. If \( a < 0 \), it opens downwards, looking like a frown.

In our specific problem, we found that \( a = 1 \), which means our parabola opens upwards. Its vertex is at \( (2, 0) \), the point where it changes direction. Using a point, such as \((4, 4)\), confirms our equation by showing another place the curve will pass through. Understanding the shape and position of a parabola is crucial because it provides insights into the behavior of quadratic functions, such as the maximum or minimum points.

General Form

Quadratic equations can also be written in general form: \( ax^2 + bx + c \). This form is derived by expanding the vertex form into a standard polynomial. It gives us another perspective on the function’s structure providing insight into roots and axis crossings.

• In this equation, \( a \) remains the leading coefficient and \( b \) and \( c \) are constants derived from expansion.

For our problem, we started with \( f(x) = (x - 2)^2 \) and expanded it to reach the general form:

• First, expand the squared binomial: \( (x - 2)^2 = x^2 - 4x + 4 \).
• Now, our equation becomes \( f(x) = x^2 - 4x + 4 \).

This is the general form of our quadratic equation. It presents a clearer view of the polynomial's degree and coefficients without focusing specifically on the vertex. This form can be more straightforward for solving using methods like factoring or the quadratic formula. Understanding both the general and vertex forms allows a comprehensive grasp of quadratic functions.