Problem 4 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The vertex of the function is \((-4, 18)\).

Step 1: Identify the Quadratic Equation

The quadratic equation given is \( f(x) = -x^2 - 8x + 2 \). This is in the standard form \( ax^2 + bx + c \) where \( a = -1 \), \( b = -8 \), and \( c = 2 \).

Step 2: Use the Vertex Formula

The vertex of a quadratic function \( f(x) = ax^2 + bx + c \) can be found using the vertex formula \( x = -\frac{b}{2a} \). Here, \( a = -1 \) and \( b = -8 \).

Step 3: Calculate the x-coordinate of the Vertex

Substitute \( b = -8 \) and \( a = -1 \) into the formula: \[ x = -\frac{-8}{2(-1)} = \frac{8}{-2} = -4 \]. The x-coordinate of the vertex is \( x = -4 \).

Step 4: Calculate the y-coordinate of the Vertex

To find the y-coordinate of the vertex, substitute \( x = -4 \) back into the function: \( f(-4) = -(-4)^2 - 8(-4) + 2 \). Simplify the expression: \[ f(-4) = -16 + 32 + 2 = 18 \].

Step 5: Conclude the Vertex Location

The vertex of the quadratic function \( f(x) = -x^2 - 8x + 2 \) is located at \((-4, 18)\).

Key Concepts

Completing the Square

Completing the square is a key algebraic technique used to transform a quadratic equation into a form where its vertex can be easily identified. Imagine you have a quadratic function in the form of \( ax^2 + bx + c \). The goal of completing the square is to rewrite this expression as \( a(x-h)^2 + k \), where \((h, k)\) represents the vertex of the parabola. Here's a quick walkthrough of how you would complete the square:

• First, factor out the coefficient \( a \) from the \( x^2 \) and \( x \) terms if \( a eq 1 \).
• Then, identify the coefficient of \( x \) (which is \( b \)), take half of it, and square the result.
• Add and subtract this square inside the parentheses, maintaining the balance of the expression.
• Simplify to acquire the vertex form, \( a(x-h)^2 + k \).

For example, in our exercise, rewriting helps us see the vertex, providing a clearer picture of the graph's symmetry and peak or valley.

Vertex Formula

The vertex formula is a straightforward method to find the vertex of a quadratic function written in standard form \( ax^2 + bx + c \). If completing the square seems complex, this formula offers a direct path:

• To find the x-coordinate of the vertex, use the formula: \( x = -\frac{b}{2a} \). This gives the central point of symmetry for the parabola.
• Substitute this \( x \)-value back into the original quadratic equation to find the y-coordinate.

Understanding this formula can significantly simplify tasks like graphing, especially under pressure in exams or quick calculations. In our original exercise, the vertex was calculated using this formula, yielding \( x = -4 \). By substituting back, the y-coordinate was determined to be \( 18 \), concluding that the vertex is \((-4, 18)\). This information is crucial for sketching the parabola accurately.

Quadratic Equation

A quadratic equation is a second-degree polynomial typically expressed as \( ax^2 + bx + c = 0 \). These equations form parabolas, which are symmetric U-shaped graphs. Key features of parabolas include:

• Their vertex, which is the highest or lowest point of the graph depending on the orientation.
• The axis of symmetry, a vertical line passing through the vertex.
• Positive or negative "a" values dictate the direction of the parabola opening: "upward" for positives and "downward" for negatives.

Their distinct shape is the basis for many practical applications, such as projectile motion and other real-world phenomena. The function in our exercise is a quadratic equation that opens downward because its leading coefficient \( a = -1 \). Knowing these parts helps you predict the graph's behavior before drawing it.

Graphing Quadratics

Graphing quadratics is a visual exercise in which you plot the parabolic path a quadratic function describes. Here's a simple outline to graph from the vertex form or after computing the vertex:

• Start by plotting the vertex on the coordinate plane. This provides a reference to form the "spine" of the graph.
• Draw a vertical line through this point for the axis of symmetry.
• Identify the direction of the parabola using the leading coefficient \( a \); it opens upward for positive \( a \) and downward for negative \( a \).
• Find additional points on either side of the axis of symmetry to shape the parabola. Often, choosing x-values that are one or two units from the vertex x-coordinate helps.
• Rechecking your calculations and symmetry can avoid missteps as you sketch.

In our exercise, graphing the quadratic \( f(x) = -x^2 - 8x + 2 \) involved plotting the vertex \((-4,18)\) and noticing its downward opening, thanks to \( a = -1 \). This visual aid is vital in understanding the full scope of quadratic behavior.