Problem 12 - Beginning and Intermediate Algebra

Question

Identify the vertex of each parabola. $$ f(x)=(x+3)^{2} $$

Step-by-Step Solution

Answer

The vertex is \(-3, 0\).

Step 1: Identify the form of the function

The given function is in the form \(f(x) = (x + 3)^2\). This is a standard form of a parabola \(f(x) = (x - h)^2 + k\) where the vertex is \((h, k)\).

Step 2: Compare the given function with standard form

In the given function \(f(x) = (x + 3)^2\), compare it with the standard form \(f(x) = (x - h)^2 + k\) to determine the values of \h\ and \k\. Here, \h = -3\ and \k = 0\.

Step 3: Determine the vertex

The vertex of the parabola \(f(x) = (x + 3)^2\) is at the point \(-3, 0\).

Key Concepts

Vertex Form

The vertex form of a parabola is very useful for identifying key features like the vertex. In its simplest terms, the vertex form of a parabola equation is written as: \(f(x) = a(x - h)^2 + k\). Here, \( (h, k) \) is the vertex of the parabola.
This form shows how the parabola is shifted from the origin. A positive or negative shift of \( h \) moves the parabola along the x-axis, while \( k \) moves it along the y-axis.
For instance, in the function \( f(x) = (x + 3)^2 \), we can see that it matches the vertex form if we rewrite it as \( f(x) = (x - (-3))^2 + 0 \).
Therefore, its vertex is at \( (-3, 0) \).
Summarizing: The vertex form helps us to quickly determine the vertex and understand the parabola's transformation.

Completing the Square

Completing the square is a technique used to manipulate a quadratic equation into its vertex form. This makes it easier to identify the vertex and graph the parabola.
To complete the square, you'll want to transform a standard quadratic equation like \( ax^2 + bx + c \) into the form \( a(x - h)^2 + k \).
Here’s a step-by-step guide:

• Start with the quadratic in its standard form.
• Isolate the x-terms by factoring out \( a \) if it's not equal to 1.
• Add and subtract the square of half the coefficient of the x-term inside the parenthesis.
• Simplify to get the vertex form.

This process makes it straightforward to extract the vertex \( (h, k) \).
For example, starting from \( x^2 + 6x + 8 \), completing the square gives us \( (x + 3)^2 - 1 \). The vertex here would be \( (-3, -1) \).
This method is a powerful tool for transforming and understanding quadratics.

Graphing Parabolas

Graphing parabolas involves plotting their key features on a coordinate plane, and the vertex is one of the most important features.
Here are the main steps for graphing a parabola:

• Identify the vertex using the vertex form \( f(x) = a(x - h)^2 + k \).
• Determine the direction of the parabola. If \( a > 0 \), it opens upwards; if \( a < 0 \), it opens downwards.
• Plot additional points by choosing x-values and calculating their corresponding y-values.
• Draw the axis of symmetry through the vertex. This is a vertical line at \( x = h \).

Graphing makes it easier to visualize the quadratic function.
For example, with the function \( f(x) = (x + 3)^2 \), the vertex is at \( (-3, 0) \), and since \( a = 1 \), it opens upwards. Plotting several points near the vertex, you can sketch the parabola's U-shape.
This approach helps in understanding the relationship between the algebraic expression and its graph.