Problem 74 - Algebra 2

Question

Graph each equation by making a table of values. \(y=-x^{2}+6 x-5\)

Step-by-Step Solution

Answer

Create a table of values, calculate corresponding \(y\) values, plot points, and draw the parabola downward.

Step 1: Understand the Equation

The equation given is a quadratic equation in the form \(y = -x^2 + 6x - 5\). Our goal is to graph this equation by using a table of values.

Step 2: Choose Values for x

Select a range of \(x\) values to substitute into the equation. Typically, values around the vertex or turning point are useful. Choose some negative and positive integers. For instance, choose \(-1, 0, 1, 2, 3, 4, 5,\) and \(6\).

Step 3: Calculate Corresponding y-values

Substitute each chosen \(x\)-value into the equation to find \(y\). For example, if \(x = 0\), then \(y = -(0)^2 + 6(0) - 5 = -5\). Repeat this for each \(x\)-value.

Step 4: Fill in the Table

Create a table to organize the \(x\) and \(y\) values. The table will have two columns: the first for \(x\) values and the second for the calculated \(y\) values. Example: \(x = 0\) yields \(y = -5\), so the table starts with \((0, -5)\).

Step 5: Plot Points on the Graph

Using the table of values, plot each \( (x, y) \) point on a coordinate plane. Start plotting from the first point and continue until all points from the table are represented on the graph.

Step 6: Sketch the Parabola

Once all points have been plotted, connect them with a smooth curve to form a parabola. The direction of the parabola will be downwards because the coefficient of \(x^2\) is negative in the equation.

Key Concepts

Graphing Quadratic Functions

Graphing a quadratic function is like drawing a beautiful U-shaped curve, called a parabola, on your graph paper. The general form of a quadratic equation is y = ax^2 + bx + c. When it comes to the graph, the value of 'a' is very important. If 'a' is positive, the parabola opens upward. If 'a' is negative, like in the equation y = -x^2 + 6x - 5, the parabola opens downward. Understanding this helps in predicting how your graph should look; whether it smiles or frowns!

Table of Values

A table of values is your best friend when graphing equations. It's used to determine points that will be plotted on your graph. To construct a table of values, follow these steps:

• Select a range of x values. These can be any numbers but choosing ones around the vertex typically gives you a well-rounded graph.
• Substitute each x value into the quadratic equation to find the corresponding y value.
• Record each x and y pair. For the given equation y = -x^2 + 6x - 5, you'd enter each calculation into the table.

Creating this table might take a little time, but it's a straightforward method to ensure you have accurate points to plot on your graph.

Vertex of a Parabola

The vertex of a parabola is like its heart. It is the point where the curve changes direction. In a quadratic equation, the vertex can be found using the formula: x = -b/(2a). For example, in the equation y = -x^2 + 6x - 5, the coefficients are 'a' = -1 and 'b' = 6. Substituting these into the formula gives the x value of the vertex: x = -6/(2*(-1)) = 3. Substitute x = 3 back into the equation to find the y-coordinate of the vertex, thus giving you the full vertex point (3, 4). When graphed, this vertex tells you the maximum point of the parabola, since this parabola opens downwards.

Coordinate Plane

The coordinate plane is a two-dimensional surface upon which we can plot points from our table of values. It consists of a horizontal axis, called the x-axis, and a vertical axis, called the y-axis. Each point on the plane is determined by an ordered pair (x, y).

• The x-value shows how far left or right the point is.
• The y-value shows how far up or down the point is.

In our exercise, once you've filled in your table of values, you'll use the coordinate plane to visualize the points and eventually draw the graph of the quadratic function. It's crucial to be precise with plotting to ensure the graph is accurate.

Plotting Points

Plotting points is the exciting step where all calculations come to life on the graph. For each ordered pair (x, y) from your table of values:

• Find the horizontal position (x) on the graph.
• Then locate the vertical position (y).
• Plot a point where these two positions meet.

After you've plotted all points from your table, you'll start to see the shape of the parabola forming. The final step is to connect these dots with a smooth curve. Remember, if the parabola opens downwards, like in our exercise, make sure the curve reflects that!