Problem 20 - College Algebra

Question

For the following exercises, determine the interval \((s)\) on which the function is increasing and decreasing. $$ f(x)=4(x+1)^{2}-5 $$

Step-by-Step Solution

Answer

Increasing on \((-1, \infty)\), decreasing on \((-\infty, -1)\).

Step 1: Identify the function type

The function given is \( f(x) = 4(x+1)^2 - 5 \). It is a quadratic function of the form \( ax^2 + bx + c \), where \( a = 4 \). Quadratic functions are parabolas, and since \( a > 0 \), the parabola opens upwards.

Step 2: Find the vertex of the quadratic

The vertex of a quadratic function \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \). For our function \( f(x) = 4(x+1)^2 - 5 \), the form is already a perfect square, so the vertex is at \( x = -1 \). This is the point where the function changes from decreasing to increasing.

Step 3: Determine intervals from the vertex

Because the parabola opens upwards, the interval where the function is increasing is to the right of the vertex and the interval where it is decreasing is to the left of the vertex. Therefore, \( f(x) \) is decreasing on \((-\infty, -1)\) and increasing on \((-1, +\infty)\).

Step 4: Write the intervals of increase and decrease

The interval where \( f(x) \) is increasing is \((-1, \infty)\), and the interval where it is decreasing is \((-\infty, -1)\).

Key Concepts

Parabolas

A parabola is a U-shaped curve that you often see in quadratic functions. In mathematics, it’s a common graph that helps illustrate quadratic equations. Quadratic functions like

• \( f(x) = ax^2 + bx + c \)
• have a distinct graph in the shape of a parabola.

The basic property of this graph is that it can open upwards or downwards. This direction depends on the coefficient \( a \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

In our exercise \( f(x) = 4(x+1)^2 - 5 \), the coefficient \( a \) equals 4, meaning the parabola opens upward.

Vertex of a Parabola

The vertex of a parabola is the point where the curve changes direction. This is also the highest or lowest point on the graph, depending on the parabola's orientation. In mathematical terms, the vertex is crucial as it defines where the function switches from increasing to decreasing or vice versa.
The vertex of a quadratic function \( ax^2 + bx + c \) is determined using the formula:

• \( x = -\frac{b}{2a} \)

In some special forms, like \( 4(x+1)^2-5 \), the equation is already squared, making it easier to spot the vertex:

• The vertex is simply at the point \( ( -1, y) \), with \( y \) being the output of the function at \( x = -1 \).

For our specific example, the vertex is located at \( x = -1 \), signifying a change from decreasing to increasing behavior.

Intervals of Increase and Decrease

Once we've identified the vertex of a parabola, we can easily determine the intervals where the function increases or decreases. The dividing line at the vertex is crucial here, as it acts as a boundary for these intervals.
For a parabola that opens upwards:

• The function will decrease on the interval to the left of the vertex.
• The function will increase to the right of the vertex.

In our given function \( f(x) = 4(x+1)^2 - 5 \), this means:

• The function decreases on the interval \((-\infty, -1)\).
• Conversely, it increases on the interval \((-1, \infty)\).

Being aware of these intervals is vital for graph analysis and functional behavior comprehension.

Upward-Opening Parabola

An upward-opening parabola resembles a smiling face. This shape indicates that the vertex is at the minimum point of the curve, adding particular properties to its behavior.The axis of symmetry runs vertically through the vertex, essentially cutting the parabola into two mirror-image halves. This characteristic symmetry comes with:

• The left side of the vertex
• (from \(-\infty \) to the vertex)
is the interval where the function decreases.
• The right side
• (from the vertex to \(\infty \))
is the interval where the function increases.

Understanding these properties is crucial for predicting a function’s behavior, especially when illustrating real-world phenomena, such as projectile motion or the shape of satellite dishes.