Problem 29 - 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. $$ (h, k)=(2,3),(x, y)=(5,12) $$

Step-by-Step Solution

Answer

The quadratic equation is \( f(x) = x^2 - 4x + 7 \).

Step 1: Write the Vertex Form of a Quadratic Equation

A quadratic function in vertex form is given by the equation \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. For this problem, substitute \((h, k) = (2, 3)\) into the equation to get: \( f(x) = a(x-2)^2 + 3 \).

Step 2: Substitute the Point into the Equation

We know the point \((x, y) = (5, 12)\) lies on the parabola. Substitute \(x = 5\) and \(y = 12\) into the equation from Step 1: \( 12 = a(5-2)^2 + 3 \).

Step 3: Solve for the Coefficient \(a\)

Simplify the equation from Step 2 to find \(a\): \( 12 = a(3)^2 + 3 \) \( 12 = 9a + 3 \) Subtract 3 from both sides: \( 9 = 9a \) Divide both sides by 9: \( a = 1 \).

Step 4: Write the Equation in Vertex Form

Now that we know \(a = 1\), substitute \(a\) back into the vertex form equation from Step 1: \( f(x) = 1(x-2)^2 + 3 \) which simplifies to: \( f(x) = (x-2)^2 + 3 \).

Step 5: Expand to Get General Form

Expand the vertex form to get the general form, \( ax^2 + bx + c \): \( f(x) = (x-2)^2 + 3 \)\( f(x) = (x^2 - 4x + 4) + 3 \)\( f(x) = x^2 - 4x + 4 + 3 \)\( f(x) = x^2 - 4x + 7 \).

Key Concepts

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is a way to express a quadratic equation that clearly shows the

• vertex of the parabola, which is a significant feature as it represents the highest or lowest point of the graph.
• This form is written as:

\[ f(x) = a(x-h)^2 + k \] where:

• \((h, k)\) is the vertex of the parabola.
• \(a\) determines the direction of the parabola (opening up if positive and down if negative) as well as the width of the parabola.

For the given example, the vertex form starts with substituting the known vertex \((h, k) = (2, 3)\). You place these values into the vertex form equation to get \( f(x) = a(x-2)^2 + 3 \). This begins the process of finding the full quadratic equation by calculating the unknown \( a \) given a point on the parabola, \((x, y) = (5, 12)\). This approach leads directly to identifying the shape and position of the parabola's curve relative to the graph.The vertex form is particularly useful for quickly sketching and understanding how changes in the variables \( a, h, \) and \( k \) affect the graph.

General Form of a Quadratic Function

The general form of a quadratic function is an alternative representation that is \[ f(x) = ax^2 + bx + c \]In this version,

• \( a, b, \) and \( c \) are constants that can be manipulated algebraically.

After determining \( a \) using the vertex form (in our case \( a = 1 \)), you can convert the vertex form equation \( f(x) = (x-2)^2 + 3 \) to the general form.Begin this transformation by expanding the squared term: \((x-2)^2 = x^2 - 4x + 4\). Add the leftover constant \(+ 3\) to finish obtaining \( f(x) = x^2 - 4x + 7 \).This form reveals the intersections of the parabola with the

• y-axis (the y-intercept ) which in this equation is the constant \( c = 7 \).

The general form is often the best choice for solving quadratic equations by factorization or using the quadratic formula, and it can be easily transformed to other forms, depending on the information needed.

Understanding Parabolas

A parabola is the graph of a quadratic function and has a U-shaped curve.This curve can either open upwards or downwards. The direction it opens is determined by the coefficient \( a \):

• If \( a > 0 \) , the parabola opens upwards.
• If \( a < 0 \) , it opens downwards.

The parabola's steepness or width is also controlled by the value of \( a \): larger values make it narrower, and smaller values make it wider.Key points about parabolas:

• The vertex is the peak or the lowest point, depending on the direction of opening.
• The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetric halves.

In the realm of quadratic functions, understanding the visual depiction of parabolas helps link the equation with its graphical features. This assists in making educated predictions and solving problems involving maximums, minimums, and intercepts. Recognizing how these attributes are influenced by altering parts of the quadratic equation is crucial for interpreting and manipulating these functions effectively in mathematical applications.