Problem 8 Make a table of \((x, y)\) value... [FREE SOLUTION]

Chapter 2: Problem 8

Make a table of \((x, y)\) values and sketch a fully labelled graph of the quadratic:\(y=-x^{2}-x+2\)

Short Answer

Expert verified

The table of values is plotted, resulting in a downward-opening parabola with vertex near \((-0.5, 2.25)\).

Step by step solution

Choose Values for x

To create a table of values, start by choosing a range of values for \(x\). In this case, let's select \(x = -3, -2, -1, 0, 1, 2, 3\).

Calculate Corresponding y Values

Substitute each \(x\) value into the quadratic equation \(y = -x^2 - x + 2\) to calculate the corresponding \(y\) values. For example, when \(x = -3\), \(y = -(-3)^2 - (-3) + 2 = -9 + 3 + 2 = -4\). Perform this calculation for each \(x\) value.

Create a Table of Values

Organize the calculated \((x, y)\) pairs into a table.\[\begin{array}{c|c}x & y \hline-3 & -4 \-2 & 0 \-1 & 2 \0 & 2 \1 & 0 \2 & -4 \3 & -10 \\end{array}\]

Sketch the Graph

Using the table of values, plot each \((x, y)\) point on a cartesian plane. Label each axis with appropriate scales. Connect the points with a smooth curve, forming a parabola, which opens downwards due to the negative sign in front of \(x^2\).

Label and Verify

Ensure that the graph is fully labeled with the equation \(y = -x^2 - x + 2\) and that the axes are correctly labeled. Verify that the shape is consistent with a parabola opening downwards and the vertex close to \((x, y) = (-0.5, 2.25)\), which can be calculated by completing the square or using vertex formula.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabola Graphing

Graphing a parabola involves plotting a curve that reveals the shape of a quadratic equation in the form of \(y = ax^2 + bx + c\). For our quadratic equation, \(y = -x^2 - x + 2\), this process starts by understanding the effect of the coefficients.

Here's a simple way to approach graphing:

Identify the leading coefficient, \(a\). A negative \(a\) means the parabola opens downwards.
Choose a set of \(x\)-values, both negative and positive, including zero, to get a symmetrical graph.
Substitute the selected \(x\)-values into the quadratic equation to find corresponding \(y\)-values.
Plot these \((x, y)\) points on a graph.

Join these points with a smooth curve. The smoother you connect them, the more accurately you will represent the shape of the parabola. Don't forget to label your axes and plot neatly for a clear presentation.

Vertex of Parabola

The vertex of a parabola is a significant point as it represents the maximum or minimum point on the graph. It is where the curve changes direction. In our equation, \(y = -x^2 - x + 2\), the parabola opens downwards, meaning the vertex is the maximum point.

To find the vertex:

Use the vertex formula \(x = -\frac{b}{2a}\), where \(b\) is the coefficient of \(x\) and \(a\) is the coefficient of \(x^2\).

In our case, \(a = -1\) and \(b = -1\), giving \(x = -\frac{-1}{2(-1)} = -\frac{1}{2}\).

Now substitute this \(x\)-value back into the original equation to find \(y\):

\(y = -(-\frac{1}{2})^2 - (-\frac{1}{2}) + 2\)
\(y = -\frac{1}{4} + \frac{1}{2} + 2 = \frac{9}{4} = 2.25\)

Thus, the vertex of the parabola is at \((-0.5, 2.25)\). Knowing the vertex helps in accurately sketching the graph, as it guides the parabola's highest or lowest point depending on the direction it opens.

Table of Values

Creating a table of values is an essential step in graphing a quadratic equation. It helps to accurately plot points that form the curve of a parabola. By selecting specific \(x\)-values and calculating the corresponding \(y\)-values, you can determine points through which the graph should pass.

Here鈥檚 how you can create and use a table of values:

Select a range of \(x\) values, ensuring some are below, above, and one is at zero for central symmetry.
For each \(x\), substitute it into the quadratic equation \(y = -x^2 - x + 2\) to get \(y\).
Record each pairing of \((x, y)\) in a table.

Our example provides a table like this:

\((-3, -4)\), \((-2, 0)\), \((-1, 2)\), \((0, 2)\), \((1, 0)\), \((2, -4)\), \((3, -10)\)

Use this table to accurately plot points on a graph. These points act as guide markers, ensuring you trace the correct shape of the parabola. The more points you can plot, the clearer and more accurate your graph will be.

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91影视

Make a table of \((x, y)\) values and sketch a fully labelled graph of the quadratic:\(y=-x^{2}-x+2\)

Short Answer

Step by step solution

Choose Values for x

Calculate Corresponding y Values

Create a Table of Values

Sketch the Graph

Label and Verify

Key Concepts

Parabola Graphing

Vertex of Parabola

Table of Values

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