Problem 30 - 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. $$ \mathbf{0}_{4}(h, k)=(-5,3),(x, y)=(2,9) $$

Step-by-Step Solution

Answer

The equation is \( y = \frac{6}{49}(x + 5)^2 + 3 \).

Step 1: Identify the Standard Form of a Parabola

The vertex form of a quadratic function is given by \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

Step 2: Substitute the Vertex Coordinates

Substitute \((h, k) = (-5, 3)\) into the vertex form equation to get \( y = a(x + 5)^2 + 3 \).

Step 3: Use the Point to Find 'a'

Substitute the point \((x, y) = (2, 9)\) into the equation to solve for \(a\). This gives \( 9 = a(2 + 5)^2 + 3 \). Simplify to find \( a \).

Step 4: Simplify the Equation

Calculate \( (2 + 5)^2 \) to get \( 49 \). Substitute it back into the equation to find \( a \): \( 9 = 49a + 3 \).

Step 5: Solve for 'a'

Subtract 3 from both sides to get \( 6 = 49a \). Divide both sides by 49 to solve for \( a \), resulting in \( a = \frac{6}{49} \).

Step 6: Write the General Form of the Equation

Substitute \( a = \frac{6}{49} \) back into the vertex form: \( y = \frac{6}{49}(x + 5)^2 + 3 \). This is the general form of the equation.

Key Concepts

Vertex Form

Quadratic functions can be represented in various ways, one of which is the vertex form. The vertex form of a quadratic function is expressed as \( y = a(x - h)^2 + k \), where

• \( (h, k) \) is the vertex, representing the "tip" or "highest point" of the parabola.
• \( a \) affects the "width" and "direction" of the parabola.

This form is beneficial because it immediately gives you the vertex of the parabola, which is important for understanding the graph's overall shape and location. An example of utilizing the vertex form is converting a given vertex \((-5, 3)\) into the equation, leading to \( y = a(x + 5)^2 + 3 \). Here, simple substitution helps form the base of our quadratic equation.

Parabolic Equations

Parabolic equations are used to describe the graphs of quadratic functions, which form U-shaped curves called parabolas. These equations can be presented in different ways, such as vertex form, standard form, and factored form. The characteristics of a parabola include:

• Vertex: The highest or lowest point on the parabola.
• Axis of symmetry: A vertical line through the vertex around which the parabola is symmetric.
• Direction: Parabolas that open upwards have a minimum point, while those that open downwards have a maximum point.
• Width: Determined by the value of \( a \); smaller \( |a| \) values result in wider parabolas.

Utilizing these characteristics helps in sketching and interpreting the graph of the quadratic function. By plugging in the vertex and solving for \( a \), a complete parabolic equation forms, giving a clear mathematical description of a graph.

Solving Quadratic Equations

Solving quadratic equations often involves finding the roots or solutions, which are the x-values where the parabola intersects the x-axis. These solutions can usually be found through:

• Factoring the quadratic when possible.
• Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Completing the square to transform the equation into vertex form.

In our exercise, the value of \( a \) was found by substituting a known point into the vertex form: \( 9 = \frac{6}{49}(x + 5)^2 + 3 \). After solving for \( a \), it became possible to fully express the quadratic equation as \( y = \frac{6}{49}(x + 5)^2 + 3 \). Understanding these methods for solving quadratic equations aids in connecting algebraic expressions to their graphical interpretations.