Problem 43 - Algebra A Combined Function

Question

Find the vertex of the graph of each function. $$ f(x)=(x+5)^{2}+2 $$

Step-by-Step Solution

Answer

The vertex is (-5, 2).

Step 1: Identify the form of the function

The given function is \(f(x) = (x+5)^2 + 2\). This function is in the vertex form of a quadratic function, which is \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex.

Step 2: Compare and find the vertex

Compare the given function \(f(x) = (x+5)^2 + 2\) with \(a(x-h)^2 + k\) to identify \(h\) and \(k\). Here, \((x+5)^2\) can be rewritten as \((x-(-5))^2\), meaning \(h = -5\) and \(k = 2\).

Step 3: State the vertex

The vertex of the quadratic function \(f(x) = (x+5)^2 + 2\) is at the point \((-5, 2)\).

Key Concepts

Quadratic Function Vertex Form

A quadratic function in vertex form is a powerful way of looking at quadratic equations. It allows us to quickly identify the vertex of the parabola, which is crucial in graphing and understanding the function's behavior. The vertex form of a quadratic function is given by: \[ f(x) = a(x-h)^2 + k \] Here, the vertex of the parabola is located at the point \( (h, k) \).

• \(a\) is a constant that affects the width and direction of the parabola (upwards if \(a > 0\) and downwards if \(a < 0\)).
• \(h\) represents the horizontal shift from the origin. If \( h = -5 \), it means the parabola shifts 5 units to the left.
• \(k\) is the vertical shift. If \( k = 2 \), the parabola shifts 2 units upwards.

This form makes it very easy to plot the main features of the graph quickly. If the quadratic equation you are dealing with is not in this form, don't worry—there's a method called completing the square to help you convert it!

Completing the Square

Completing the square is a mathematical technique used to convert a standard form quadratic equation into its vertex form. Let's break it down into simple steps. First, let's consider a quadratic equation in the standard form: \[ ax^2 + bx + c \] The goal is to rearrange this into the vertex form, \( a(x-h)^2 + k \). Here's how:

• Start with the quadratic and linear terms, \( ax^2 + bx \). If \(a\) is not 1, factor \(a\) out of these two terms.
• Find the perfect square trinomial by adding and subtracting \( \left( \frac{b}{2} \right)^2 \) inside the equation. Remember to balance this equation.
• Rearrange into \( (x - h)^2 \) and simplify.

This technique efficiently finds the vertex form, which beautifully reveals the graph's vertex. It's a handy tool especially for solving equations graphically or analytically.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, which generally take the shape of a parabola when graphed. These are written in the format: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants. Here are the essential aspects of quadratic equations:

• The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of \(a\).
• Quadratic equations can have either two real solutions, one real solution, or no real solutions depending on the discriminant, \(b^2 - 4ac\).
• The vertex of the parabola is crucial because it represents the maximum or minimum point of the function, depending on the direction of the parabola.

Understanding quadratic equations is essential in algebra as they appear in various applications, from physics to engineering. The ability to convert from standard form to vertex form using methods like completing the square gives you more flexibility in analyzing and solving them.