Problem 30 - 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. $$ (h, k)=(-5,3),(x, y)=(2,9) $$

Step-by-Step Solution

Answer

The general form of the quadratic equation is \( f(x) = \frac{6}{49}x^2 + \frac{60}{49}x + \frac{297}{49} \).

Step 1: Understand the Vertex Form

The vertex form of a quadratic function is given by \[ f(x) = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola. In this exercise, the vertex is given as \((-5, 3)\).

Step 2: Substitute the Vertex into the Vertex Form

Substitute \(h = -5\) and \(k = 3\) into the equation:\[ f(x) = a(x + 5)^2 + 3 \] Note that \(x - (-5) = x + 5\).

Step 3: Use the Point to Solve for \(a\)

Use the given point \((2, 9)\) to find \(a\). Substitute \(x = 2\) and \(f(x) = 9\) into the equation:\[ 9 = a(2 + 5)^2 + 3 \]Simplify and solve for \(a\):\[ 9 = a(7)^2 + 3 \]\[ 9 = 49a + 3 \]\[ 6 = 49a \]\[ a = \frac{6}{49} \]

Step 4: Write the Equation in Vertex Form

Substitute \(a = \frac{6}{49}\) back into the vertex form equation:\[ f(x) = \frac{6}{49}(x + 5)^2 + 3 \]

Step 5: Convert to General Form

To convert into the general form, expand \((x + 5)^2\) and distribute \(a\):\[ f(x) = \frac{6}{49}(x^2 + 10x + 25) + 3 \]\[ f(x) = \frac{6}{49}x^2 + \frac{60}{49}x + \frac{150}{49} + 3 \]Convert \(3\) to a fraction with the same denominator:\[ f(x) = \frac{6}{49}x^2 + \frac{60}{49}x + \frac{150}{49} + \frac{147}{49} \]Combine the constants:\[ f(x) = \frac{6}{49}x^2 + \frac{60}{49}x + \frac{297}{49} \]

Step 6: Final Verification Step

Recheck calculations and ensure correct simplification to verify that the general form was correctly derived from the vertex form, accommodating every arithmetic operation accurately.

Key Concepts

Vertex Form

The vertex form of a quadratic function is an important way to express parabolas, especially when you want to know the vertex, or the "turning point" of the graph. The formula for vertex form is \[ f(x) = a(x - h)^2 + k \] where

• \( a \) is the coefficient that affects the direction and the width of the parabola,
• \( (h, k) \) is the vertex, meaning the highest or lowest point on the graph.

By looking at the vertex form, it's easy to see where the parabola "opens up" or "goes down". If \( a \) is positive, the parabola opens upwards. If \( a \) is negative, it opens downwards. Converting general quadratic equations to this form can reveal insights about the graph's shape and position.

General Form

The general form is often how quadratic equations are initially presented. The general quadratic function is represented as \[ f(x) = ax^2 + bx + c \] where

• \( a \) determines the parabola's opening direction and width,
• \( b \) influences the parabola's axis and vertex,
• \( c \) is the y-intercept, the point where the graph meets the y-axis.

Converting from the vertex to the general form involves expanding and simplifying the equation, as shown in the solution steps. This form is often used when solving quadratic equations using methods like factoring, completing the square, or using the quadratic formula.

Solving for a

To find the specific value of \( a \) in the function, we substitute a given point on the parabola into the equation. This requires one known point, \((x, y)\), and the vertex \((h, k)\). In this exercise, using the point \((2, 9)\), we substitute into the equation:1. Start with vertex form: \( f(x) = a(x - h)^2 + k \).2. Replace terms with known values: \( 9 = a(2 + 5)^2 + 3 \).3. Solve the equation: \[ 9 = 49a + 3 \].4. Isolate \( a \): \[ a = \frac{6}{49} \].This method shows that by using specific points, we can determine how much the parabola "stretches" and "compresses" around its vertex.

Parabola Vertex

The vertex of a parabola is a key point as it signifies its highest or lowest point, based on the direction it opens. Given the vertex, you can determine much about the parabola's nature and position in the plane. For example, in our problem, the vertex is \((-5, 3)\). It indicates that:

• The parabola reaches its peak or bottom at point \((-5, 3)\),
• The axis of symmetry runs vertically through \(x = -5\),
• \( k = 3 \) sets the y-value of the vertex directly.

Using the vertex is crucial when writing quadratic equations in vertex form. This not only identifies the graph's centerline of reflection but also helps in understanding transformations such as translations or shifts from the standard parabola \( f(x) = x^2 \).