Problem 14 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. \(g(x)=(x-3)^{2}\)

Step-by-Step Solution

Answer

The graph is a symmetric parabola with vertex at (3, 0).

Step 1: Choose x-values

To graph the function, start by selecting a range of x-values. A good range might be from -2 to 8 to capture the behavior around the vertex.

Step 2: Calculate Corresponding y-values

Use the function \(g(x) = (x-3)^2\) to calculate the y-values for your chosen x-values. Perform the calculations as follows:\[\begin{align*}g(-2) &= (-2-3)^2 = 25,\g(-1) &= (-1-3)^2 = 16,\g(0) &= (0-3)^2 = 9,\g(1) &= (1-3)^2 = 4,\g(2) &= (2-3)^2 = 1,\g(3) &= (3-3)^2 = 0,\g(4) &= (4-3)^2 = 1,\g(5) &= (5-3)^2 = 4,\g(6) &= (6-3)^2 = 9,\g(7) &= (7-3)^2 = 16,\g(8) &= (8-3)^2 = 25.\end{align*}\]

Step 3: Create the Table of Values

List the x-values and their corresponding y-values in a table:\[\begin{array}{c|c}x & g(x) \\hline-2 & 25 \-1 & 16 \0 & 9 \1 & 4 \2 & 1 \3 & 0 \4 & 1 \5 & 4 \6 & 9 \7 & 16 \8 & 25 \\end{array}\]

Step 4: Plot the Points

On a graph, plot each pair of \( (x, g(x)) \) points from the table. For example, plot the following points: \((-2, 25), (-1, 16), (0, 9), (1, 4), (2, 1), (3, 0), (4, 1), (5, 4), (6, 9), (7, 16), (8, 25)\).

Step 5: Draw the Graph

Connect the plotted points smoothly to reveal the parabola. The vertex of the parabola is at \(x = 3\) and \(g(3) = 0\), and you'll see the graph is symmetrical around this point with arms extending upwards.

Key Concepts

Graphing Parabolas

Graphing parabolas can be straightforward once you understand the basic structure of this type of function. A parabola is the graph of a quadratic function, which is typically in the form of - \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The simplest form, like our function \(g(x) = (x-3)^2\), actually comes in the shape \(y = (x-h)^2 + k\). This is called vertex form,- where \(h\) and \(k\) dictate the parabola's vertex. To graph a parabola, you first need to create a collection of points that lie on the function. - These are determined using x-values input into the function to obtain y-values. You then- plot these coordinate points - and draw a smooth curve through them to visualize the parabola. Remember:

• Parabolas are symmetrical. Whatever you calculate on one side of the vertex mirrors on the other side.
• The direction the parabola opens (upwards or downwards) is defined by the sign of \(a\). For example, if \(a\) is positive, it opens upwards."

Vertex of a Parabola

The vertex of a parabola is a crucial point. It represents either the highest point in a parabola that opens downward or the lowest point for one that opens upward. In the vertex form of a quadratic function, which is \(y = (x-h)^2 + k\), the vertex is simply the point \((h, k)\).- For the function \(g(x) = (x-3)^2\), the parabola's vertex is at the point \((3, 0)\).To locate the vertex:

• Identify the value of \(h\) that shifts the graph left or right.
• Determine the value of \(k\) to understand how far up or down the graph is shifted.

Once these values are pinpointed, they give the axis of symmetry. Any changes in values must adhere to this axis.Recognizing the vertex's location aids in understanding the overall shape and trajectory of the parabola.

Table of Values

Creating a table of values is a helpful step in graphing a quadratic function like a parabola. This process involves selecting a set of x-values and computing the corresponding y-values using the function equation.For instance, using the function \(g(x) = (x-3)^2\), you can select x-values such as -2, -1, 0, 1, 2, 3, etc.For each x-value chosen, substitute it into the function to find the y-value,- giving a pair \((x, g(x))\). These pairs form coordinate points to be plotted on a graph. Examples include:

• \((-2, 25)\)
• \((3, 0)\)
• \((8, 25)\)

This table helps in visualizing the parabola's path and confirms the shape being symmetrical around the vertex.Plotting these on a graph reveals the continuous "U"-shape characteristic of parabolas. Having many plotted points ensures accuracy and ease in drawing the curve precisely.