Graph The Line: Fill In A T-Chart For Y = X²

Graph The Line1 Fill In A T Chart To Graph Y X2 Fill In A T Chart

Graph The Line1 Fill In A T Chart To Graph Y X2 Fill In A T Chart

Perform the following tasks related to graphing linear equations, including filling in T-charts, identifying slopes and intercepts, converting to slope-intercept form, and graphing lines with given slopes and points. Use these instructions to guide your understanding of the methods used in graphing linear functions and solving related problems.

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Graphing linear equations is a fundamental skill in algebra, enabling the visualization of relationships between variables and understanding the behavior of lines. This exercise involves creating T-charts, determining slopes, identifying intercepts, converting equations to slope-intercept form, and graphing lines with specified slopes and points. Mastery of these concepts enhances problem-solving proficiency and conceptual understanding of linear functions.

Constructing T-Charts for Graphing

To graph the equations, start by creating T-charts that record pairs of (x, y) values. For the equation y = x, choose a range of x-values, such as -2, -1, 0, 1, 2, then compute corresponding y-values, which are equal to x in this case, resulting in pairs: (-2, -2), (-1, -1), (0, 0), (1, 1), (2, 2). Plotting these points yields a diagonal line with a slope of 1.

For y = -2x - 3, select x-values such as -2, -1, 0, 1, 2. Calculate y-values: for x = -2, y = -2(-2) - 3 = 4 - 3 = 1; for x = -1, y = -2(-1) - 3 = 2 - 3 = -1; for x = 0, y = -3; for x = 1, y = -2(1) - 3 = -2 - 3 = -5; for x = 2, y = -4 - 3 = -7. The pairs are (-2, 1), (-1, -1), (0, -3), (1, -5), (2, -7). Plotting these points produces a line with a slope of -2 and a y-intercept at -3.

Similarly, for y = -x^3 - 3x, pick x-values and compute y accordingly, or for y = -x^3 + 9x. These involve cubic functions; choosing x-values like -2, -1, 0, 1, 2, and calculating y will help visualize the curves. For example, for x = 1 in y = -x^3 + 9x, y = -1 + 9 = 8; for x = -1, y = -(-1)^3 + 9(-1) = 1 - 9 = -8.

Determining the Slope of a Line

The slope of a line between two points (x₁, y₁) and (x₂, y₂) is calculated as:

slope = (y₂ - y₁) / (x₂ - x₁)

For example, between points (-5, -4) and (4, ?), and (-4, 2) and (3, ?), substitute the known points to find the slopes:

Between (-5, -4) and (4, y), if y is unknown, more information is needed; otherwise, if both points are known, compute the difference in y divided by the difference in x.

Between (-4, 2) and (3, -), assuming the second y-coordinate is provided, say, y = -x - 1, then the slope between (-4, 2) and (3, -5) (since y = -x - 1, at x=3, y=-4) can be calculated accordingly.

Identifying Equations in Slope-Intercept Form

An equation y = mx + b is in slope-intercept form, where m is the slope, and b is the y-intercept. For equations like y = -x + 1, the slope is -1, and the intercept is at (0, 1).

Similarly, y = x + 1 has a slope of 1 and a y-intercept at (0, 1).

For equations not initially in slope-intercept form, such as y + 5 = -5(x + 3), rearrange to get y by itself: y = -5x - 15 - 5, which simplifies to y = -5x - 20.

Converting to Slope-Intercept Form and Graphing

To write equations in slope-intercept form, isolate y on one side. For example, y + 5 = -5(x + x - 3), expand and simplify:

y + 5 = -5x - 5x + 15, then y = -5x - 5x + 15 - 5, which simplifies to y = -10x + 10.

Graphting lines with given slopes and points involves plotting the y-intercept and using the slope to find additional points. For example, a line with slope 2 passing through (-6, _) can be graphed by starting at a point on the y-axis and applying the slope ratio to extend the line.

An undefined slope indicates a vertical line passing through a given x-coordinate, such as x = -3,1 in this case, which is a vertical line through the points (-3, 1) and potentially others with the same x-value.

Summary and Application

Understanding how to fill T-charts, compute slopes, identify intercepts, and convert equations facilitates the graphing process and deepens comprehension of linear functions. Whether working with simple linear equations like y = x, y = -2x - 3, or more complex cubic functions, these foundational skills are essential for successful graphing and analysis in algebra and calculus.

References

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