Problem 13 - Algebra and Trigonometry

Question

Graph each equation in Exercises \(13-28 .\) Let \(x=-3,-2,-1,0\) \(1,2,\) and 3. $$y=x^{2}-2$$

Step-by-Step Solution

Answer

The graph of the function \(y=x^{2}-2\) is a parabola opening upwards with its vertex shifted down by 2 units. It passes through the points satisfying the equation for each value of x in the set \(-3, -2, -1, 0, 1, 2, 3\).

Step 1: Calculate y for each x

To find the y coordinate for each value of x, substitute the value of x into the equation \(y=x^{2}-2\). Do this for each of the given x values.

Step 2: Plot the points

After determining the coordinate pairs \((x,y)\), plot these points on your graph.

Step 3: Draw the Graph

Connect the points on your graph with a smooth curve. This graph should be a parabola shape opening upwards with its vertex shifted down by 2 units.

Key Concepts

Parabola

A parabola is a U-shaped curve that you often see when graphing quadratic equations. When you graph the equation \( y = x^2 - 2 \), you are plotting a parabola. The basic shape of a parabola is always the same, but it can be stretched, compressed, or moved on the graph.

Let's break down what makes a graph a parabola:

• It can open either upwards or downwards. In this case, it opens upwards.
• Its symmetry is its main feature—it looks the same on both sides of its vertex.
• The vertex is the "tip" of the parabola or the point where it turns around.

When graphing, always look for the vertex first as it is crucial in defining the shape and position of the parabola.

Coordinates

In graphing any equation, coordinates play a crucial role. A coordinate is a pair of numbers that define the position of a point on a graph. Usually written as \((x, y)\), these pairs reveal the exact location of a point.

For the equation \(y = x^2 - 2\), your task is to calculate and plot the coordinates given the specific x-values. Each coordinate pair comes from substituting an x-value into the equation.

• For example, if \(x = -3\), then \(y = (-3)^2 - 2 = 7\). So, the coordinate pair is \((-3, 7)\).
• This process repeats for all x-values like \(x = -2, -1, 0, 1, 2,\) and \(3\).

By plotting these calculated points, you can piece together the shape of the parabola.

Vertex of a Quadratic Function

When talking about quadratics, the vertex is one of the most important points on the graph of a parabola. It is the highest or lowest point, depending on whether the parabola opens upwards or downwards. Here, because \(y = x^2 - 2\) opens upwards, the vertex is the lowest point.

To find the vertex, remember the general form for a quadratic equation, \(y = ax^2 + bx + c\). For our equation, \(a = 1, b = 0,\) and \(c = -2\), so it can be written as \(y = 1x^2 + 0x - 2\). The vertex can be calculated using the formula for the x-coordinate, which is \(x = \frac{-b}{2a}\). Here the vertex x-value is \(x = 0\).

Substituting \(x = 0\) back into the equation, the y-value of the vertex is \(-2\), so the vertex is

• \((0, -2)\)

Finding the vertex first helps in sketching the parabola more accurately.