Problem 48 - College Algebra with Modeling and Visualization

Question

Use transformations to sketch a graph of \(f\). \(f(x)=(x+4)^{2}\)

Step-by-Step Solution

Answer

Shift the parabola \(y = x^2\) 4 units left.

Step 1: Identify the Base Function

The base function here is \(f(x) = x^2\), which is a parabola that opens upwards with its vertex at the origin (0,0).

Step 2: Identify the Transformation

The given function is \(f(x) = (x + 4)^2\). This involves a horizontal shift. Specifically, the \(+4\) inside the parenthesis means the base function \(x^2\) is shifted left by 4 units to \(x = -4\).

Step 3: Sketch the Transformed Graph

Start by plotting the vertex of the parabola at \((-4, 0)\), which results from shifting the original vertex \((0, 0)\) 4 units to the left. The shape of the parabola remains the same, opening upwards. Draw the parabola symmetrically around the vertex \((-4, 0)\).

Key Concepts

Quadratic Functions

Quadratic functions are one of the fundamental concepts in algebra and appear frequently in mathematics. A quadratic function is represented by the equation \( f(x) = ax^2 + bx + c \).
This equation represents a curve which is a parabola. The most basic form of a quadratic function is \( f(x) = x^2 \).
Here, the term \( ax^2 \) dominates the equation, especially when \( x \) is large. This specific term gives the curve its distinctive parabolic shape, curving away in a 'U' shape when the coefficient \( a \) is positive and an inverted 'U' shape when \( a \) is negative.

• If \( b \) or \( c \) are non-zero, they modify the position of the parabola on the graph.
• The vertex form of a quadratic function, \( f(x) = a(x-h)^2 + k \), highlights its peak point or the vertex \((h,k)\), making it easy to graph transformations like shifts and stretches.

Understanding quadratic functions is crucial because they model many natural phenomena, from projectile motion to the design of satellite dishes.

Parabolas

Parabolas are the graphical representation of quadratic functions and have a symmetrical, curved shape. They can open upwards or downwards, depending on the sign of the leading coefficient \(a\) in the quadratic equation \(f(x) = ax^2 + bx + c\).
When \(a > 0\), the parabola opens upwards, resembling a 'U'. When \(a < 0\), it opens downwards, looking like an upside-down 'U'.
Every parabola has a vertex, which is its highest or lowest point, depending on its orientation. This vertex is a key feature and can be the point where the curve changes direction.
The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.

• The vertex in the standard form \(ax^2 + bx + c\) can be found using \(x = -\frac{b}{2a}\).
• The vertex form, \(f(x) = a(x-h)^2 + k\), explicitly shows the vertex as \((h, k)\).
• Parabolas are used in various applications like satellite dish designs and suspension bridge architectures due to their focusing properties and geometric characteristics.

Recognizing the basic components of parabolas helps in understanding more complex mathematical transformations and properties.

Horizontal Shifts

Horizontal shifts are a crucial transformation in graph plotting that change the position of functions on a graph without altering their shape. When we talk about horizontal shifts, we are moving the entire graph left or right.
This type of transformation is evident in the function \(f(x) = (x + c)^2\), where \(c\) affects the direction and magnitude of the shift.

• If \(c > 0\), the graph shifts \(c\) units to the left. For example, \( (x + 4)^2 \) shifts four units left.
• If \(c < 0\), the graph shifts \(|c|\) units to the right.

Horizontal shifts do not affect the shape of the graph or its orientation,
they just reposition its location in respect to the axes.
Understanding horizontal shifts is essential for graph transformations, as they are often used in combination with other transformations like vertical shifts or stretches/compressions to graph complex functions with ease.