Problem 56 - Beginning and Intermediate Algebra

Question

If the following transformations are performed on the graph of \(f(x)\) to obtain the graph of \(g(x),\) write the equation of \(g(x)\). \(f(x)=x^{2}\) is shifted right 5 units and down 1.5 units.

Step-by-Step Solution

Answer

\(g(x) = (x - 5)^2 - 1.5\)

Step 1: Shift the graph right 5 units

When the graph is shifted to the right by 5 units, we will replace \(x\) with \((x-5)\) in the equation \(f(x)\). This is because, for any point on the original graph, when its \(x\)-coordinate is increased by 5, the corresponding function value (i.e., \(y\)-coordinate) remains the same. So, we have: \(f_a(x) = ((x-5))^2\)

Step 2: Shift the graph down 1.5 units

Now that we have the equation of the graph shifted right 5 units (\(f_a(x)\)), we will apply the second transformation, i.e., shifting it down by 1.5 units. For this, we will subtract 1.5 from the function value (i.e., \(y\)-coordinate). The equation of \(g(x)\) is: \(g(x) = f_a(x) - 1.5\) Now, substitute the expression for \(f_a(x)\) from Step 1: \(g(x) = ((x-5))^2 - 1.5\) The equation of \(g(x)\) after applying both transformations to the graph of \(f(x)\) is: \(g(x) = (x - 5)^2 - 1.5\)

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function where the highest exponent of the variable is 2. The general form of a quadratic function is expressed as \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The most basic form of a quadratic function is \(f(x) = x^2\), which is called a parabola. It is a symmetric curve with its vertex at the origin, \((0, 0)\).

• When \(a > 0\), the parabola opens upwards. If \(a < 0\), it opens downwards.
• The vertex form of a quadratic function is \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola.
• Quadratic functions are widely used in various fields such as physics (for modeling projectile motion), finance, and optimization problems.

Understanding the structure of a quadratic function is crucial for performing graph transformations and identifying the shifts and changes applied to the graph.

Horizontal Shift

Horizontal shifts involve moving the graph of a function left or right along the x-axis. For a quadratic function like \(f(x) = x^2\), a horizontal shift is achieved by replacing \(x\) with \(x - h\) in the function's equation, where \(h\) represents the number of units the graph shifts.

• If the graph moves to the right, replace \(x\) with \(x - h\).
• If it shifts to the left, replace \(x\) with \(x + h\).

In our original exercise, the graph of \(f(x) = x^2\) is shifted 5 units to the right. This is applied by substituting \(x\) with \(x-5\), resulting in the equation \(f_a(x) = (x-5)^2\). This transformation does not alter the shape of the graph, only its position along the x-axis.

Vertical Shift

Vertical shifts shift the entire graph of a function up or down along the y-axis. For a quadratic function \(f(x) = x^2\), a vertical shift can be achieved by adding or subtracting a constant \(k\) to the function's result:

• To move the graph up, add \(k\) to the function: \(f(x) = x^2 + k\)
• To move it down, subtract \(k\): \(f(x) = x^2 - k\)

In our problem, after shifting horizontally, the function is then moved down by 1.5 units. This is achieved by subtracting 1.5 from the function obtained after the horizontal shift, leading to \(g(x) = (x-5)^2 - 1.5\). These types of shifts are straightforward and do not change the "steepness" or orientation of the parabola, but simply raise or lower the entire graph.

Function Transformations

Function transformations involve changing the graph of a function through various operations including shifts, stretches, compressions, and reflections.

• Shifts: Move the graph left, right, up, or down without altering its shape.
• Stretches and Compressions: Alter the width or height of the graph, achieved by multiplying the function by a factor greater or less than one respectively.
• Reflections: Flip the graph over an axis (x or y) by multiplying by \(-1\).

In the context of the original exercise, we have applied two simple transformations: a horizontal shift and a vertical shift, resulting in the function \(g(x) = (x-5)^2 - 1.5\). These transformations provide a powerful method for tailoring graphs to meet specific criteria or fit particular data sets. Understanding how each transformation impacts the graph helps us manipulate quadratic functions with ease, achieving desired graph configurations effectively.