Problem 43 - College Algebra Essentials

Question

Graph the given functions, \(f\) and \(g,\) in the same rectangular coordinate system. Select integers for \(x\), starting with \(-2\) and ending with \(2 .\) Once you have obtained your graphs, describe how the graph of g is related to the graph of \(f\). $$ f(x)=x^{2}, g(x)=x^{2}+1 $$

Step-by-Step Solution

Answer

The function \(g(x) = x^{2}+1\) is the graph of the function \(f(x) = x^{2}\) shifted upwards by 1 unit.

Step 1: Graphing \(f(x)=x^{2}\)

Start with the function \(f(x) = x^{2}\). Plug in integers from -2 to 2 as \(x\) values into the function and find corresponding \(y\) values. These will be: (-2,4), (-1,1), (0,0), (1,1), and (2,4). Plot these points on the graph and draw the curve.

Step 2: Graphing \(g(x)=x^{2}+1\)

Move on to the next function \(g(x)=x^{2}+1\). Similarly, Plugin integers from -2 to 2 as \(x\) values into the function and find corresponding \(y\) values. These will be: (-2,5), (-1,2), (0,1), (1,2), and (2,5). Plot these points on the same graph and draw the curve.

Step 3: Comparing Both Graphs

Both functions have the same parabolic shape since they are both quadratic functions. The graph of \(g(x)\) appears to be a vertical translation of the graph of \(f(x)\) as it is shifted upward by 1 unit. Therefore, the function \(g(x)\) is the graph of \(f(x)\) shifted upwards by 1 unit.

Key Concepts

Parabolic Graphs

When graphing quadratic functions, such as the function f(x) = x^2, the resultant graph is known as a parabola. This specific type of curve is symmetrical and opens either upwards, like a traditional 'U' shape, or downwards, resembling an upside-down 'U', depending on the coefficient of the squared term. In the case of f(x) = x^2, this produces a right-side-up 'U' shape because the coefficient is positive.

For a basic understanding, imagine plotting points for various x values and then drawing a smooth curve to connect these points. For example, plotting the points (-2,4), (-1,1), (0,0), (1,1), and (2,4) results in a symmetrical graph about the y-axis, which is a characteristic feature of the parabolic graph of a quadratic function where the squared term has no coefficient other than 1.

Vertical Translation

Vertical translation in the context of graphing functions refers to shifting the entire graph of a function up or down without changing its shape. With quadratic functions like our example g(x) = x^2 + 1, the '+1' represents a vertical translation of the basic x^2 parabola one unit higher on the y-axis.

It's crucial to understand that vertical translation does not affect the width, orientation, or the symmetry of the original graph; it simply relocates the graph. Therefore, if you were to take the graph of f(x) = x^2, and move it up by one unit, you'd achieve the graph of g(x). This property allows us to quickly sketch graphs of functions that are vertical translations without plotting individual points, saving time and effort.

Quadratic Equations

Quadratic equations are mathematical expressions of the form ax^2 + bx + c = 0, where a, b, and c are numerical coefficients and a is not zero. These equations are so named because the highest power of the variable x is two, termed as 'quadratic', which comes from the Latin word 'quadratum' for square.

Every quadratic equation corresponds to the graph of a quadratic function, and it can often be solved algebraically using methods such as factoring, completing the square, or applying the quadratic formula. Parabolas are the graphical representation of these equations on the coordinate plane. In-depth comprehension of how quadratic equations form parabolas, and the ability to manipulate and understand their translations and transformations, is crucial in both algebraic problem-solving and graphical analysis.