Problem 55 - Precalculus with Limits

Question

In Exercises 55-62, write an equation for the function that is described by the given characteristics. The shape of \(f(x) = x^2\), but shifted three units to the right and seven units downward

Step-by-Step Solution

Answer

The function that matches the given description is \(g(x) = (x-3)^2-7\).

Step 1: Identify Function Transformations

The function needs to be shifted three units to the right, which is achieved by replacing \(x\) with \(x-3\) in the equation. Further, it needs to be moved seven units down; so, \(f(x)\) will be replaced with \(f(x)-7\). The given function is \(f(x) = x^2\), so the transformations will lead to the function \(g(x) = (x-3)^2-7\).

Step 2: Write Down the Final Equation

After applying the transformations, the equation of the function becomes \(g(x) = (x-3)^2-7\)

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function with the general form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). It is called 'quadratic' because it involves squaring the variable \(x\). This function produces a U-shaped graph known as a "parabola". Parabolas can open upward or downward:

• They open upward when the coefficient \(a\) is positive.
• They open downward when \(a\) is negative.

The most basic quadratic function is \(f(x) = x^2\), where the vertex of the parabola is at the origin (0,0), and the axis of symmetry is the y-axis. The shape and orientation of the parabola are directly influenced by the coefficients in the function.

Horizontal Shift

Horizontal shifts are transformations that move a function left or right on the graph. For a function \(f(x)\), a horizontal shift is caused by replacing \(x\) with \(x + k\) or \(x - k\):

• Replacing \(x\) with \(x - k\) moves the graph to the right by \(k\) units.
• Replacing \(x\) with \(x + k\) moves the graph to the left by \(k\) units.

For instance, in the transformation of \(f(x) = x^2\) to \(g(x) = (x-3)^2\), you replace \(x\) with \(x-3\). This shifts the parabola three units to the right. The vertex of the parabola moves from (0,0) to (3,0), reflecting this horizontal shift.

Vertical Shift

Vertical shifts involve moving the entire graph of a function up or down. These transformations are typically executed by changing the y-values of the function by adding or subtracting a constant \(c\):

• Adding \(c\) to the function, \(f(x) + c\), shifts it upwards by \(c\) units.
• Subtracting \(c\), \(f(x) - c\), shifts it downwards by \(c\) units.

In the provided exercise, the shift involves moving \(f(x) = x^2\) seven units downward, which modifies the function to \(g(x) = (x-3)^2 - 7\). This subtraction of 7 moves the entire parabola down, changing the vertex from (3,0) to (3,-7) and effectively lowering each point on the graph by 7 units.