Problem 12 - Technical Mathematics with Calculus

Question

Graph each function. Set the viewing window for \(x\) and \(y\) initially from -5 to 5 then resize if needed. $$x^{2}-2 x-y+2=0$$

Step-by-Step Solution

Answer

The graph is a parabola with vertex at (1, 1), opening upwards. The initial window range (-5,5) for both x and y is suitable for viewing the entire parabola.

Step 1: Rearrange the equation to solve for y

To graph the function, we need to solve for y. Start by adding y to both sides of the equation and then move the remaining terms to the other side to get y by itself: \[y = x^{2} - 2x + 2\]

Step 2: Analyze the equation

The equation is now in the form \[y = ax^2 + bx + c\]which represents a parabola that opens upwards because the coefficient of the squared term, (a), is positive.

Step 3: Find the vertex of the parabola

The vertex of a parabola in the form \[y = ax^2 + bx + c\] is found using the formula \[h = -\frac{b}{2a}, k = \frac{4ac - b^2}{4a}\]For our equation, \[a = 1, b = -2, c = 2\]Thus, \[h = -(-2)/(2*1) = 1\]\[k = (4*1*2 - (-2)^2)/(4*1) = (8 - 4)/4 = 1\]The vertex is at (1, 1).

Step 4: Determine the axis of symmetry

The axis of symmetry can be found from the vertex and is the line \[x = h\]In this case, the axis of symmetry is \[x = 1\].

Step 5: Plot the vertex and axis of symmetry

Plot the point (1,1) on the graph as the vertex of the parabola. Then draw a dotted vertical line through x=1 as the axis of symmetry.

Step 6: Find additional points

Choose x-values around the vertex to find additional points for the parabola. For example, use x=0, x=2 (which are one unit away from the vertex at x=1). Plug these x-values into the equation to find corresponding y-values.

Step 7: Graph the parabola

Using the vertex, axis of symmetry, and the additional points found, sketch the parabola. Make sure it opens upward, is symmetrical with respect to the axis of symmetry, and passes through all the calculated points.

Step 8: Adjust the viewing window if necessary

Initially, set the viewing window from -5 to 5 for both x and y. Since the vertex is at (1,1), this window should be sufficient. If the parabola is not fully visible, adjust the window accordingly to capture the shape of the parabola fully.

Key Concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is essential for graphing parabolas, as these equations often represent the foundation of a parabola's shape. A quadratic equation can usually be identified by its standard form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero.

When solving for \( y \), the goal is to get \( y \) on one side of the equation by itself. This often involves moving terms across the equality sign and simplifying. For the given exercise, we initially have \( x^2 - 2x - y + 2 = 0 \). Through simple algebraic manipulation, adding \( y \) to both sides and rearranging terms gives us \( y = x^2 - 2x + 2 \).

This format is now much more conducive for graphing as it presents \( y \) as a function of \( x \) and highlights the quadratic nature of the parabola.

Finding the Vertex of a Parabola

The vertex of a parabola represents the turning point where the graph changes direction. For a parabola that opens up or down, the vertex corresponds to the minimum or maximum value of the function, respectively.

To find the vertex of a parabola represented by \( y = ax^2 + bx + c \), we use the vertex formula \( h = -\frac{b}{2a} \) and \( k = \frac{4ac - b^2}{4a} \). In our exercise, we have \( a = 1 \) and \( b = -2 \) and \( c = 2 \). By substituting these values into the vertex formula, we get the vertex \( (1, 1) \).

This is a key step in graphing the parabola as we can center our graph around this point, ensuring we capture the most significant feature of the parabolic curve.

Axis of Symmetry

The axis of symmetry is a straight line that divides the parabola into two mirror-image halves. For any parabola in the form of \( y = ax^2 + bx + c \), the axis of symmetry can be directly derived from the vertex's \( h \) value as \( x = h \).

From our earlier calculation, we found the vertex to be at (1, 1), meaning the axis of symmetry is the line \( x = 1 \). When graphing the parabola, we draw a dotted vertical line through this x-value to denote the axis of symmetry. This tool not only aids in plotting the parabola correctly but also ensures that the two halves of the parabola are congruent, reflecting the inherent symmetry of the shape.

Parabola Graph Analysis

Identifying Key Features

After determining the vertex and axis of symmetry for a parabola, we proceed to examine its graph. Identifying key features includes locating the vertex, as mentioned, and plotting points symmetrically on both sides of the axis.

Plotting Additional Points

To ensure accuracy, we select x-values around the vertex—both greater and lesser—and solve for corresponding y-values. These additional points provide us with more information to sketch the parabola's curve accurately.

Assessing the Viewing Window

When graphing, it's helpful to start with a standard viewing window, such as from -5 to 5 for both \( x \) and \( y \) axes. Visualizing the plotted points, we can then extend or shrink this window to ensure the entire curve is visible, capturing the essence of the parabola's graph. This process not only makes the graph more detailed but also more meaningful for analysis.