Question

THOUGHT PROVOKING Write and graph a transformation of the graph of \(f(x)=x^5-3 x^4+2 x-4\) that results in a graph with a \(y\)-intercept of \(-2\).

Short answer

The transformed function is \(g(x)=x^5-3x^4+2x-2\), which is a vertical shift 2 units upwards of the original function \(f(x)=x^5-3x^4+2x-4\). The graph of the transformed function will now have a y-intercept at -2.

Step-by-step solution

Calculate Original y-Intercept

To find the y-intercept of the original function, substitute \(x=0\) in \(f(x)=x^5-3x^4+2x-4\). This gives \(f(0)=-4\). So, the original y-intercept is -4.

Calculate Required Shift

The desired y-intercept is -2. Subtract the original y-intercept from the desired y-intercept to find how much the function needs to be shifted vertically. This gives \(-2 - (-4) = 2\). So, a shift of 2 units upward is required.

Write Transformed Function

To achieve a vertical shift of 2 units upwards, 2 should be added to the original function. This gives the transformed function as \(g(x)=x^5-3x^4+2x-2\).

Graph Original and Transformed Functions

Plot both the original function \(f(x)=x^5-3x^4+2x-4\) and the transformed function \(g(x)=x^5-3x^4+2x-2\) on the same coordinate plan. You will see that the graph of the transformed function is a vertical shift 2 units upwards of the original function graph.

Key concepts

Polynomial Functions

Understanding polynomial functions is fundamental to studying algebra and calculus. A polynomial function is a mathematical expression consisting of variables and coefficients, which involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial function is

\(f(x) = x^5 - 3x^4 + 2x - 4\)

This particular function is a 5th-degree polynomial because the highest exponent of the variable \(x\) is 5. The coefficients in this case are the numbers in front of the variable terms: 1 for \(x^5\), -3 for \(x^4\), and 2 for \(x\). The constant term is -4. The degree of the polynomial indicates the shape and the number of turns the graph might take.

Vertical Shifts

A vertical shift in a graph is described as moving the whole graph up or down without changing its shape. It is a key concept in function transformation where you add or subtract a constant to the output value \(y\) of the function.

For instance, if you want to shift the graph of \(f(x) = x^5 - 3x^4 + 2x - 4\) up by 2 units, you simply add 2 to the entire function:

\(g(x) = f(x) + 2 = x^5 - 3x^4 + 2x - 4 + 2\)

The new function \(g(x)\) represents the vertically shifted graph. Vertical shifts are crucial for adjusting functions to fit certain criteria, such as changing the y-intercept.

Y-Intercept Calculations

The y-intercept is the point where a graph crosses the y-axis, which occurs when the input \(x\) is zero. Calculating the y-intercept is straightforward for polynomial functions; you substitute \(x\) with 0 and solve for \(y\).

For example, with the function \(f(x) = x^5 - 3x^4 + 2x - 4\), the y-intercept is found by calculating \(f(0)\):

\(f(0) = (0)^5 - 3(0)^4 + 2(0) - 4 = -4\)

The y-intercept here is -4, represented as the point (0, -4) on the coordinate plane. Accurate calculation of y-intercepts is important when graphing functions or when the y-intercept itself is a significant attribute of the graph, such as in the case of this exercise.

Function Transformation

Function transformation involves changing the appearance or position of a graph in a systematic way. The primary types of transformations include shifting, stretching, compressing, and reflecting. Each type of transformation changes the graph in a different manner while maintaining its basic shape.

For the case of polynomial function transformation, when the exercise requires changing the y-intercept, a vertical shift is performed. Adjusting the original function \(f(x)\) by adding or subtracting a value gives us a new function. In this scenario, we calculated the y-intercept of the polynomial \(f(x) = x^5 - 3x^4 + 2x - 4\) as -4 and aimed for a new y-intercept of -2. To do so, we shifted the graph vertically by adding the difference to each \(y\) value of the original function, which resulted in the new function:

\(g(x) = x^5 - 3x^4 + 2x - 2\)

It's essential to grasp these concepts thoroughly to perform accurate function transformations to meet specific criteria.