Problem 14 - Algebra Form and Function

Question

The graphs are horizontal and/or vertical shifts of the graph of \(y=x^{2}\). Find a formula for each function graphed.

Step-by-Step Solution

Answer

Answer: To determine the equation of the graph, follow these steps: 1. Identify the general equation for vertical and horizontal shifts: \(y = (x-h)^2 + k\). 2. Determine the values of \(h\) and \(k\) for the given graph by observing the vertex of the parabola. 3. Substitute the values of \(h\) and \(k\) into the general equation to obtain the formula for the shifted function.

Step 1: Identify the general equation for vertical and horizontal shifts

In order to find a formula for a function that has been shifted horizontally and/or vertically, we must first recall the general equation for such a function. Given a function \(y = f(x)\), if the graph is shifted horizontally by \(h\) units and vertically by \(k\) units, the equation for the new function will be: \(y = f(x-h)+k\) For our specific case, since the original function is \(y = x^2\), the equation for the shifted function will be: \(y = (x-h)^2 + k\)

Step 2: Determine the values of h and k

Our task is to find the values of \(h\) and \(k\) that correspond to the given function graphs. In general, we can determine the direction of the shift by observing the vertex of the parabola (i.e., the point at which the graph changes its direction) and compare its position in the original function \((0,0)\) with its position in the shifted function \((h,k)\). To find the specific horizontal and vertical shifts that apply to each specific graph you are working with, follow these two steps: 1. Determine the horizontal shift (h): Find the vertex of the given graph and compare it with the vertex of the original function. The difference in their x-coordinates indicates the horizontal shift, with a positive value corresponding to a shift to the right and a negative value corresponding to a shift to the left. 2. Determine the vertical shift (k): The difference in y-coordinates between the vertices of the original function and the given graph indicates the vertical shift, with a positive value corresponding to a shift upward and a negative value corresponding to a shift downward.

Step 3: Substitute the values of h and k into the general equation

Once you have found the values of \(h\) and \(k\) for the given graph, substitute them back into the general equation \(y = (x-h)^2 + k\). This will give you the formula for the shifted function. In summary, the steps to find the formula for a function graph that is a horizontal and/or vertical shift of the graph of \(y = x^2\) are: 1. Identify the general equation for vertical and horizontal shifts: \(y = (x-h)^2 + k\). 2. Determine the values of \(h\) and \(k\) for the given graph by observing the vertex of the parabola. 3. Substitute the values of \(h\) and \(k\) into the general equation to obtain the formula for the shifted function. Please repeat steps 2 and 3 for each graph given in the exercise.

Key Concepts

Quadratic Functions

Quadratic functions form the foundation of many mathematical concepts and are expressed through polynomial equations of the form \(f(x) = ax^2 + bx + c\). These equations generate a curved graph known as a parabola. The term 'quadratic' itself refers to the highest degree of the variable \(x\) which, in these functions, is always squared.

Key properties of quadratic functions include:

• They open either upwards or downwards depending on the sign of the coefficient \(a\). If \(a\) is positive, the parabola opens upwards, while a negative \(a\) causes it to open downwards.
• The vertex of the parabola represents the maximum or minimum point of the graph, depending on whether the parabola opens downwards or upwards.
• The axis of symmetry is a vertical line that passes through the vertex, effectively dividing the parabola into two mirror-image halves.

Quadratic functions not only help model various real-world phenomena, such as projectile motion and area calculation, but they also form the basis for more complex algebraic equations.

Vertex Form

Vertex form is a specific way of expressing quadratic functions that makes graphing them straightforward, especially when focusing on shifts. This form is represented as \(y = a(x-h)^2 + k\). Here:

• \((h, k)\) are the coordinates of the vertex, making it easy to locate the parabola's pivotal point on a graph.
• \(h\) determines how the graph shifts horizontally. A positive \(h\) value moves the parabola to the right, while a negative \(h\) shifts it left.
• \(k\) accounts for vertical shifts. It elevates the graph when positive and lowers it when negative.

Using vertex form simplifies the process of identifying changes in a parabola's orientation and placement. This form is particularly beneficial when determining the impact of each transformation, as each component of the expression directly corresponds to specific shifts in the graph.

Parabola Graphing

Graphing a parabola involves plotting its shape on a coordinate plane based on its equation, particularly when expressed in vertex form. The steps involved are simple but crucial for accuracy:

• Start by locating the vertex, \((h, k)\), which guides the graph's starting point.
• Identify the direction of the parabola. The coefficient \(a\) in the equation \(y = a(x-h)^2 + k\) determines whether it opens upwards or downwards.
• Calculate and plot additional points for accuracy. Since the parabola is symmetrical around the vertex, determine several points on one side and mirror them across the axis of symmetry.
• Finally, connect these points smoothly to illustrate the curved shape of the parabola.

Graphing a parabola in this method highlights its key features, allowing better understanding of its behavior and transformations. Whether you're studying its practical uses or investigating theoretical principles, mastering parabola graphing is an essential skill in algebra.