Problem 14 - Precalculus Essentials

Question

Graph each equation.Let \(x=-3,-2,-1,0\) \(1,2,\) and 3 $$y=x^{2}+2$$

Step-by-Step Solution

Answer

The coordinates (-3,11), (-2,6), (-1,3), (0,2), (1,3), (2,6), and (3,11) can be plotted to represent the equation \( y = x^{2} + 2 \).

Step 1: Identify x-values

The given x-values are -3, -2, -1, 0, 1, 2 and 3.

Step 2: Substitution and solve for y

Substitute each x-value into the equation \( y = x^{2} + 2 \) to solve for \( y \). For instance, if \( x = -3 \), then \( y = (-3)^{2} + 2 = 9 + 2 = 11 \). Repeat this for all the x-values: when \( x = -2\), \(y = 6\); when \(x = -1\), \(y = 3\); when \(x = 0\), \(y = 2\); when \(x = 1\), \(y = 3\); when \(x = 2\), \(y = 6\); and when \(x = 3\), \(y = 11\).

Step 3: Plot the graph

The generated coordinates from step 2 are (-3,11), (-2,6), (-1,3), (0,2), (1,3), (2,6), and (3,11). These can then be plotted on a graph using an appropriate scale.

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function characterized by an equation of the form \( y = ax^2 + bx + c \). In the context of this exercise, we are working with the quadratic function \( y = x^2 + 2 \). This particular function is in standard form where \( a = 1 \), \( b = 0 \), and \( c = 2 \).
Important features of quadratic functions include their parabolic shape when graphed. This means they either open upwards or downwards, depending on the sign of \( a \). Since \( a \) is positive in our case, the parabola opens upwards.

• The vertex is the point where the parabola changes direction. Given the formula, the vertex is at \((0, c)\) for the simplest form \(y = x^2 + c\).
• The axis of symmetry is a vertical line that passes through the vertex, and for our function, it is \(x = 0\).

Understanding these characteristics helps in sketching the graph and predicting the behavior of the quadratic function.

Plotting Points

When plotting points for graphing equations, the first step is to calculate the output (or \( y \) value) for each chosen input (or \( x \) value). In this problem, we have been provided with a range of \( x \) values: \(-3, -2, -1, 0, 1, 2,\) and \(3\).
For each \( x \), you substitute it into the quadratic function \( y = x^2 + 2 \) to find the corresponding \( y \) value. For example:

• When \( x = -3 \), \( y = 9 + 2 = 11 \)
• When \( x = 0 \), \( y = 0 + 2 = 2 \)
• When \( x = 3 \), \( y = 9 + 2 = 11 \)

These calculations yield the coordinates (-3,11), (-2,6), (-1,3), (0,2), (1,3), (2,6), (3,11). Each pair \((x, y)\) represents a point that can be plotted on the coordinate plane.

Coordinate Plane

The coordinate plane is a two-dimensional surface used for graphing where each point is determined by an ordered pair \((x, y)\). The horizontal axis is called the \(x\)-axis, and the vertical axis is the \(y\)-axis.
When graphing quadratic functions, like \( y = x^2 + 2 \), it is crucial to select an appropriate scale for the axes. This ensures that all points are clearly visible and the shape of the graph is accurately captured.

• Start by marking the \(x\)-axis with the values provided, in this case, \(-3\) through \(3\).
• Then, use the calculated \( y \) values to determine how to label your \( y \)-axis, ensuring it includes the range of \( y \) values from the calculations, here \(2\) to \(11\).

Plot each point you've calculated and draw a smooth curve that passes through all the points. The parabola should be symmetrical with respect to the vertex.

Substitution Method

The substitution method in mathematics is a simple process of replacing variables with numbers to make calculations easier. When dealing with functions, it involves plugging specific \( x \) values into the function to find the corresponding \( y \) values.
In our exercise, you substitute each \( x \)-value into the function \( y = x^2 + 2 \). For example:

• If \( x = 1 \), substitute into the equation: \( y = 1^2 + 2 \)
• This gives \( y = 1 + 2 = 3 \)

Repeat this for all given \( x \) values to create a complete set of points to plot. This method not only helps in finding solutions but also in understanding how the input affects the output in the structure of the function. It is a foundational tool for drawing the graph of any function, especially quadratic ones.