Problem 40 - Algebra A Combined Function

Question

Find the vertex of the graph of each function. $$ f(x)=(x-3)^{2} $$

Step-by-Step Solution

Answer

The vertex of the function \( f(x) = (x-3)^2 \) is \( (3, 0) \).

Step 1: Identify the form of the quadratic function

The given function is expressed as \( f(x) = (x - 3)^2 \). This is a quadratic function in vertex form, which is \( f(x) = (x - h)^2 + k \), where \( (h, k) \) is the vertex.

Step 2: Determine the vertex from the function

In the equation \( f(x) = (x - 3)^2 \), compare it to the general vertex form \( f(x) = (x - h)^2 + k \). Here, \( h = 3 \) and \( k = 0 \) since there is no \( + k \) term. Hence, the vertex of the parabola is \( (3, 0) \).

Step 3: Confirm the vertex location

Given the completed square form \( (x-3)^2 + 0 \), the term \( (x-3) \) indicates a horizontal shift 3 units to the right on the graph. This confirms that \( h = 3 \) and with no vertical shift \( k = 0 \), the vertex remains at \( (3, 0) \).

Key Concepts

Quadratic Function

Quadratic functions are a type of polynomial function characterized by their highest degree term being squared. They form a parabolic shape when graphed on a coordinate plane. A standard quadratic function appears as \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants and \( a eq 0 \). Quadratics are unique in that they can open upwards or downwards:

• If \( a > 0 \), the parabola opens upwards and the vertex is the minimum point.
• If \( a < 0 \), the parabola opens downwards and the vertex is the maximum point.

Remember, the vertex is a critical point where the function changes direction, and this helps determine the function's graph.

Vertex Form

The vertex form of a quadratic function is a powerful way of expressing quadratics, which directly shows the vertex of the parabola. The general form is \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. This form is particularly useful for quickly identifying transformations:

• The term \((x-h)^2\) indicates a horizontal shift \( h \) units from the origin.
• The constant \( k \) reflects vertical shifts.

In our example \( f(x) = (x-3)^2 \), the vertex form reveals the vertex precisely, making it easy to spot without additional calculations. Also, this form simplifies sketching the graph of the quadratic.

Parabola Vertex

The vertex of a parabola is the point where it turns. It is either the highest or lowest point, depending on the direction of the parabola. For a quadratic in vertex form \( f(x) = a(x-h)^2 + k \), the vertex is given directly by the point \((h, k)\). Understanding the parabola's vertex is essential for analyzing the graph's behavior:

• Horizontal shifts are represented by \( h \).
• Vertical shifts and transforms are given by \( k \).

In our example, since \( f(x) = (x-3)^2 \), the vertex is at \( (3, 0) \), meaning the graph moves 3 units right and remains on the x-axis, showcasing a simple parabola shape that opens upwards.

Completing the Square

Completing the square is a technique used to transform a quadratic function into its vertex form. It's especially handy when you start with the standard form \( ax^2 + bx + c \) and need to identify the vertex quickly. This method involves:

• Rearranging the equation such that \( ax^2 + bx \) is the focus.
• Adding and subtracting a specific value to make a perfect square trinomial.

For instance, if we had a quadratic \( f(x) = x^2 - 6x + 9 \), completing the square involves transforming the expression into \( (x-3)^2 \) by understanding how each part affects the graph. While our current function is already in vertex form, knowing how to complete the square helps us understand and derive vertex positions from standard quadratic forms efficiently.