Problem 8 - Algebra A Combined Function

Question

Find the vertex of the graph of each quadratic function by completing the square or using the vertex formula. $$ f(x)=x^{2}-9 x+8 $$

Step-by-Step Solution

Answer

The vertex is \( \left( \frac{9}{2}, -\frac{49}{4} \right) \).

Step 1: Identify Quadratic Function

We are given the quadratic function \( f(x) = x^2 - 9x + 8 \). The goal is to find the vertex of its graph.

Step 2: Recognize the Standard Form

The function \( f(x) = ax^2 + bx + c \) is in standard form, where \( a = 1 \), \( b = -9 \), and \( c = 8 \).

Step 3: Use the Vertex Formula

The vertex \((h, k)\) of a quadratic function in standard form can be found using the vertex formula: \( h = -\frac{b}{2a} \).

Step 4: Calculate Vertex's x-coordinate

Substitute \( b = -9 \) and \( a = 1 \) into the formula to calculate \( h \). \[ h = -\frac{-9}{2 \times 1} = \frac{9}{2} \]

Step 5: Calculate Vertex's y-coordinate

Substitute \( x = \frac{9}{2} \) into the function to find \( k \):\[ k = \left(\frac{9}{2}\right)^2 - 9 \left(\frac{9}{2}\right) + 8 = \frac{81}{4} - \frac{81}{2} + 8 = \frac{81}{4} - \frac{162}{4} + \frac{32}{4} = -\frac{49}{4}.\]

Step 6: Write the Vertex Solution

The vertex of the function \( f(x) = x^2 - 9x + 8 \) is \( \left( \frac{9}{2}, -\frac{49}{4} \right) \).

Key Concepts

Completing the Square

Completing the square is a method used in algebra to transform a quadratic equation into a perfect square trinomial. This makes it easier to find the vertex of a quadratic function and solve for the variable. To complete the square, we take a quadratic expression of the form \( ax^2 + bx + c \) and rewrite it into the form \( a(x-h)^2 + k \), where \((h, k)\) will be the vertex.
Here is how it works:

• Divide the entire quadratic expression by \( a \) if \( a eq 1 \).
• Take half of the coefficient \( b \), square it, and add and subtract it inside the function.
• Factor the new trinomial into a square, which gives \( (x-h)^2 \).

This method is particularly useful because it directly reveals the vertex \((h, k)\). By practicing completing the square, understanding and visualizing the graph of quadratic functions becomes much easier.

Vertex Formula

The vertex formula is a quick and efficient way to find the vertex of a quadratic equation in the form \( ax^2 + bx + c \). This formula helps you find the \( x \)-coordinate of the vertex without completing the square.
The formula is:

• \( h = -\frac{b}{2a} \)

Once you find \( h \), substitute it back into the original function to find \( k \), which represents the \( y \)-coordinate of the vertex:

• \( k = f(h) \)

For the equation \( f(x) = x^2 - 9x + 8 \), using the vertex formula simplifies the process significantly. It confirms the vertex \( \left( \frac{9}{2}, -\frac{49}{4} \right) \). Understanding this formula is crucial as it provides a direct pathway to find vertex points on a graph.

Standard Form of Quadratic Functions

The standard form of a quadratic function is crucial as it presents the function in the most common format. In the context of quadratic equations, it is given by:

• \( ax^2 + bx + c \)

where \( a \), \( b \), and \( c \) are constants. This form is helpful because it showcases the key components necessary for various methods, such as completing the square or applying the vertex formula. It serves as a starting point for many operations.
In our example, the quadratic function \( f(x) = x^2 - 9x + 8 \) is already in the standard form with \( a = 1 \), \( b = -9 \), and \( c = 8 \), allowing us to execute these methods efficiently.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree. Their general form is \( ax^2 + bx + c = 0 \). They represent parabolas when graphed on a coordinate plane, where the direction of the parabola (upward or downward) is determined by the sign of \( a \).
Quadratic equations can be solved by several methods:

• Factoring, if the equation can be simplified into two binomials.
• Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which provides the roots of the equation.
• Completing the square, to find real or complex solutions.

Understanding quadratic equations extends beyond finding roots; it's also about analyzing their graphs and properties, such as the vertex, axis of symmetry, and whether they open upwards or downwards.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a field of mathematics that describe geometric shapes in algebraic terms. It is essential for understanding the behavior of quadratic functions by presenting them as graphs on a coordinate plane.
When we graph quadratic functions, we typically plot a parabola. Key features of this parabola include:

• The vertex, which is the highest or lowest point.
• The axis of symmetry, a vertical line that passes through the vertex.
• The direction of opening, which depends on the coefficient \( a \).

By using coordinate geometry, students can visualize algebraic equations and see how transformations affect graphs. It bridges the gap between abstract equations and their spatial representations, helping to foster a deeper comprehension of the quadratic relationships.