Graphing y = x + 2

Graph the equation $y = x + 2$ using a table of values.

Choose x-values: Select convenient values: $x = -2, -1, 0, 1, 2$ = Five x-values chosen

Calculate y for x = -2: $y = (-2) + 2 = 0$ = Point: $(-2, 0)$

Calculate y for x = -1: $y = (-1) + 2 = 1$ = Point: $(-1, 1)$

Calculate y for x = 0: $y = (0) + 2 = 2$ = Point: $(0, 2)$

Calculate y for x = 1: $y = (1) + 2 = 3$ = Point: $(1, 3)$

Calculate y for x = 2: $y = (2) + 2 = 4$ = Point: $(2, 4)$

Plot and connect: Plot all five points and draw a straight line through them = Line with slope 1, y-intercept 2

Answer: The graph is a straight line passing through $(-2, 0)$, $(-1, 1)$, $(0, 2)$, $(1, 3)$, and $(2, 4)$.

Graphing y = -2x + 3

Graph the equation $y = -2x + 3$ using the table method.

Choose x-values: Use $x = -1, 0, 1, 2, 3$ = Five x-values selected

Calculate y for x = -1: $y = -2(-1) + 3 = 2 + 3 = 5$ = Point: $(-1, 5)$

Calculate y for x = 0: $y = -2(0) + 3 = 0 + 3 = 3$ = Point: $(0, 3)$

Calculate y for x = 1: $y = -2(1) + 3 = -2 + 3 = 1$ = Point: $(1, 1)$

Calculate y for x = 2: $y = -2(2) + 3 = -4 + 3 = -1$ = Point: $(2, -1)$

Calculate y for x = 3: $y = -2(3) + 3 = -6 + 3 = -3$ = Point: $(3, -3)$

Plot and connect: Plot points and draw a line that goes down from left to right = Decreasing line with slope $-2$

Answer: The graph is a decreasing line (slopes downward) passing through $(0, 3)$ with slope $-2$.

Graphing y = (1/2)x - 1

Graph $y = \frac{1}{2}x - 1$ using a table of values.

Choose x-values wisely: Use even numbers to avoid fractions: $x = -4, -2, 0, 2, 4$ = Smart choice of x-values

Calculate y for x = -4: $y = \frac{1}{2}(-4) - 1 = -2 - 1 = -3$ = Point: $(-4, -3)$

Calculate y for x = -2: $y = \frac{1}{2}(-2) - 1 = -1 - 1 = -2$ = Point: $(-2, -2)$

Calculate y for x = 0: $y = \frac{1}{2}(0) - 1 = 0 - 1 = -1$ = Point: $(0, -1)$

Calculate y for x = 2: $y = \frac{1}{2}(2) - 1 = 1 - 1 = 0$ = Point: $(2, 0)$

Calculate y for x = 4: $y = \frac{1}{2}(4) - 1 = 2 - 1 = 1$ = Point: $(4, 1)$

Plot and connect: Plot all points and draw a line with gentle upward slope = Line with slope $\frac{1}{2}$, y-intercept $-1$

Answer: The graph passes through $(0, -1)$ and rises gently (slope $\frac{1}{2}$) from left to right.

Mistake: Forgetting to use parentheses when substituting negative values

Why: For $y = 2x + 1$ with $x = -3$: writing $2 \times -3 + 1$ instead of $2(-3) + 1$ can lead to sign errors.

Correct: Always write $2(-3) + 1 = -6 + 1 = -5$. The parentheses ensure correct multiplication.

Mistake: Switching x and y when plotting points

Why: The point $(3, -2)$ means $x = 3$ (horizontal) and $y = -2$ (vertical). Swapping gives the wrong location.

Correct: Remember: x comes first (go right/left), then y (go up/down). Plot $(3, -2)$ by going right 3, then down 2.

Mistake: Not using enough points to confirm the line

Why: Two points determine a line, but calculation errors might not be caught.

Correct: Use at least 3-5 points. If one doesn't fall on the line with the others, check your calculation for that point.

Mistake: Drawing a curved line through the points

Why: Linear equations ALWAYS produce straight lines. If your points seem curved, there's an error.

Correct: If points don't form a straight line, recalculate each y-value. For linear equations, the graph is always a straight line.

Cell Phone Data Plans

Phone companies charge a base fee plus a rate per gigabyte. Graphing helps compare plans.

A plan costs 20 euros per month plus 5 euros per GB. The equation is $y = 5x + 20$ where $x$ is GB used and $y$ is total cost.

Temperature Conversion

Converting between Celsius and Fahrenheit follows a linear equation that can be graphed.

The formula $F = \frac{9}{5}C + 32$ converts Celsius to Fahrenheit. Graphing shows how temperatures relate.

Q: How many points do I need to graph a line?

A: Technically, two points determine a line. However, using 3-5 points is recommended because it helps you catch calculation errors. If one point doesn't align with the others, you know to recheck your work.

Q: Which x-values should I choose?

A: Choose values that are easy to calculate. Include negative numbers, zero, and positive numbers for a complete picture. If the equation has fractions, pick multiples of the denominator to get whole number results.

Q: What if my points don't form a straight line?

A: If you're graphing a linear equation (like $y = mx + b$) and your points don't line up, there's a calculation error. Go back and check each substitution, paying special attention to negative numbers and order of operations.