Problem 13 - Beginning and Intermediate Algebra

Question

Identify the vertex of each parabola. $$ f(x)=(x+3)^{2}-4 $$

Step-by-Step Solution

Answer

The vertex is \( (-3, -4) \).

Step 1 - Identify the form of the quadratic equation

The given function is in the form of a transformed standard parabola equation. The standard form is \[ f(x) = (x-h)^2 + k \].

Step 2 - Identify the values of h and k

In the equation \[ f(x) = (x + 3)^2 - 4 \], compare it to the standard form \[ f(x) = (x-h)^2 + k \] to identify h and k. Note that \( (x + 3)^2 \) can be rewritten as \( (x - (-3))^2 \). Hence, \( h = -3 \). The term \( -4 \) represents \( k \). Thus, \( k = -4 \).

Step 3 - Determine the vertex

The vertex of a parabola in the form \[ (x-h)^2 + k \] is given by the point \( (h, k) \). Using the values \( h = -3 \) and \( k = -4 \), the vertex is \( (-3, -4) \).

Key Concepts

quadratic equation

A quadratic equation represents a polynomial of degree 2. In other words, it includes terms with the variable raised to an exponent of 2 as the highest degree. The general form of a quadratic equation is: \[ ax^2 + bx + c = 0 \] where:

• \(a\) is the coefficient of \(x^2\) (cannot be zero)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

Knowing this form helps you to understand and manipulate these equations more easily, whether you are solving for roots, factoring, or analyzing the properties of their graphs. Quadratic equations often produce a parabola when graphed on a coordinate plane.

standard form

The standard form of a quadratic function makes it easy to identify key characteristics, such as the vertex. The standard form of a quadratic equation is given by: \[ f(x) = a(x-h)^2 + k \] where:

• \(a\) determines the width and direction of the parabola
• \(h\) and \(k\) represent the coordinates of the vertex of the parabola

In this form, \(h\) is derived by taking the opposite sign of the number inside the parentheses with \(x\), and \(k\) remains as it is in the equation. For example, in \[ f(x) = (x+3)^2 - 4 \] by comparing it with \[ f(x) = (x-h)^2 + k \] we find that \(h = -3\) and \(k = -4\). This informs us directly where the vertex of the parabola is located. Hence, the vertex is \( (-3, -4) \).

transformation of functions

Transformations of functions assist in understanding how to shift, reflect, and scale graphs. For quadratic functions, certain transformations can make determining the graph’s properties easier. The common transformations include:

• **Horizontal Shifts**: In \[ (x-h)^2 \], \(h\) shifts the graph left or right. \(h > 0\) shifts the graph to the right, and \(h < 0 \) shifts it to the left.
• **Vertical Shifts**: Adding or subtracting a constant \(k\) outside the square term \[ (x-h)^2 + k \] will shift the graph up or down. \(k > 0\) moves the graph upwards, and \(k < 0\) moves it downwards.
• **Reflections**: If the coefficient \(a\) in \[ a(x-h)^2 + k \] is negative, it reflects the graph across the x-axis, flipping it upside down.
• **Vertical Stretches/Shrinks**: The value of \(a\) will stretch the graph vertically if\( |a| > 1 \) and shrink it if \( |a| < 1 \).

Combining these transformations helps graph the quadratic and locate its vertex. For instance, identifying \( h and k \) in \[ f(x)=(x+3)^2-4 \] allows us to see that the parabola shifts 3 units left and 4 units down, giving a vertex at \( (-3, -4) \).