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

Step-by-Step Solution

Answer

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

Step 1: Understand the Vertex Form

The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. In this problem, the vertex is \((2, 3)\), which means \( h = 2 \) and \( k = 3 \).

Step 2: Substitute Vertex into the Vertex Form

Plug the vertex \((h, k) = (2, 3)\) into the vertex form: \[ f(x) = a(x - 2)^2 + 3 \]. This equation has one unknown parameter, \(a\), that we need to find.

Step 3: Use the Given Point to Find \(a\)

We are given a point \((x, y) = (5, 12)\) that lies on the parabola. Substitute \(x = 5\) and \(y = 12\) into the equation: \[ 12 = a(5 - 2)^2 + 3 \].

Step 4: Solve for \(a\)

Simplify and solve \(12 = a(3)^2 + 3\) to find \(a\):1. Compute \( (3)^2 = 9\), so the equation becomes \(12 = 9a + 3\).2. Subtract 3 from both sides to get \(9 = 9a\).3. Divide by 9: \(a = 1\).

Step 5: Write the Quadratic Equation in Standard Form

Substitute \(a = 1\) back into the vertex form equation: \[ f(x) = (x - 2)^2 + 3 \]. Expand this to write it in the standard form:1. Compute \((x - 2)^2 = x^2 - 4x + 4\).2. Add 3 to get \(f(x) = x^2 - 4x + 7\).Thus, the standard form of the quadratic equation is \(f(x) = x^2 - 4x + 7\).

Key Concepts

Vertex Form

The vertex form of a quadratic function is a useful way to express the equation of a parabola. It takes the form \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) represents the vertex of the parabola. This format easily shows the vertex, which is the turning point of the parabola and is either the maximum or minimum point, depending on the orientation of the function.

If the parabola opens upwards, the vertex is a minimum. Conversely, if it opens downward, the vertex is a maximum.

For our exercise, the vertex is given as \((2, 3)\), meaning our equation starts as \( f(x) = a(x - 2)^2 + 3 \). The only missing piece here is the "\(a\)" value, which determines the width and direction (upwards or downwards) of the parabola.

To find "\(a\)", we use an additional point on the curve.

Quadratic Function

A quadratic function is a type of polynomial function. It is characterized by its highest degree term being \(x^2\). Its general shape is that of a parabola, which can either open upwards or downwards based on its leading coefficient.

Quadratic functions are given in the standard form \( ax^2 + bx + c \) or in other forms like the vertex form or intercept form, which provide specific insights into the properties of the graph.

In our example, the quadratic function equation comes from substituting a point \((x, y)\) into the vertex form to solve for "\(a\)". By knowing \(a\), as well as \(h\) and \(k\) from the vertex, we can derive the exact quadratic function that represents our parabola.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is \( ax^2 + bx + c \). It is useful for analyzing the overall shape and intercepts of the parabola.

This form doesn't directly reveal the vertex like the vertex form does, but it provides easy access to the y-intercept, which is the point where the parabola crosses the y-axis \((0, c)\).

Transforming an equation from vertex form to standard form involves expanding the squared term \((x - h)^2\) and then combining any constant terms. For example, starting with \( (x - 2)^2 + 3 \), expanding gives \( x^2 - 4x + 4 \), and adding 3 results in \( x^2 - 4x + 7 \), which is the desired standard form of the equation.

This transformation facilitates further analysis of the quadratic function, such as finding roots or understanding the symmetry of the graph.