Representations of Functions: Matching a table to a function

To determine the corresponding equation for the given table, follow these steps:

• Step 1: Confirm the linearity of the data by calculating the slope, $m$, using consecutive data points:
  - Between points $(-2, 0)$ and $(0, 1)$, the slope is $m = \frac{1 - 0}{0 - (-2)} = \frac{1}{2}$.
  - Between points $(0, 1)$ and $(2, 2)$, the slope is $m = \frac{2 - 1}{2 - 0} = \frac{1}{2}$.
  - Confirm the same slope $m = \frac{1}{2}$ for remaining pairs $(2, 2)$ and $(4, 3)$, $(4, 3)$ and $(6, 4)$.
• Step 2: Use one point, such as $(0, 1)$, to find the y-intercept, $b$, knowing the slope $m = \frac{1}{2}$:
  - Use the slope-intercept form: $y = mx + b$. Substituting $(0, 1)$, gives $1 = \frac{1}{2}(0) + b$, implying $b = 1$.
• Step 3: Formulate the equation: Given $m = \frac{1}{2}$ and $b = 1$, the linear function is:
  - $y = \frac{1}{2}x + 1$.
• Step 4: Compare with provided choices:

The equation $y = \frac{1}{2}x + 1$ matches choice 4. Therefore, the correct equation corresponding to the table is $y = \frac{1}{2}x + 1$.

To solve this problem, we'll follow these steps:

• Step 1: Determine the form of the potential equation.
• Step 2: Calculate the slope using two points from the table.
• Step 3: Identify the y-intercept.

Now, let's work through each step:

Step 1: Determine the Equation Form
Since the relationship appears linear, we'll use $y = mx + b$.

Step 2: Calculate the Slope
Using the points $(-1, 1)$ and $(0, 2)$, calculate the slope:
$m = \frac{2 - 1}{0 - (-1)} = \frac{1}{1} = 1$.

Step 3: Identify the Y-Intercept
Using the slope $m = 1$ and point $(0, 2)$, since the y-intercept $b$ is the $y$-value when $x = 0$, $b = 2$.

Thus, the equation is $y = x + 2$.

Verify by checking all points from the table:
- For $X = -1, Y = 1$: $1 = -1 + 2$
- For $X = 0, Y = 2$: $2 = 0 + 2$
- For $X = 1, Y = 3$: $3 = 1 + 2$
- For $X = 2, Y = 4$: $4 = 2 + 2$
- For $X = 3, Y = 5$: $5 = 3 + 2$

Thus the equation $y = x + 2$ satisfies all table values.

Therefore, the solution to the problem is $y = x + 2$.

We will begin by using the formula for finding slope:

$m=\frac{y_2-y_1}{x_2-x_1}$

First let's take the points:

$(-1,4),(3,8)$

$m=\frac{8-4}{3-(-1)}=$

$\frac{8-4}{3+1}=$

$\frac{4}{4}=1$

Next we'll substitute the point and slope into the line equation:

$y=mx+b$

$8=1\times3+b$

$8=3+b$

Lastly we'll combine like terms:

$8-3=b$

$5=b$

Therefore, the equation will be:

$y=x+5$

To determine which equation corresponds to the function given by the table, we will test each equation using the $(x, y)$ pairs from the table. Specifically, we will verify which equation satisfies all pairs so that we can conclude it functions as desired.

Consider the equation $y = \frac{2}{3}x + 2\frac{2}{3}$.

• For $x = -1$:

Substitute into the equation:
$y = \frac{2}{3}(-1) + 2\frac{2}{3} = -\frac{2}{3} + \frac{8}{3} = \frac{6}{3} = 2$. The value matches the table, $y = 2$.

• For $x = 2$:

Substitute into the equation:
$y = \frac{2}{3}(2) + 2\frac{2}{3} = \frac{4}{3} + \frac{8}{3} = \frac{12}{3} = 4$. The value matches the table, $y = 4$.

• For $x = 5$:

Substitute into the equation:
$y = \frac{2}{3}(5) + 2\frac{2}{3} = \frac{10}{3} + \frac{8}{3} = \frac{18}{3} = 6$. The value matches the table, $y = 6$.

• For $x = 8$:

Substitute into the equation:
$y = \frac{2}{3}(8) + 2\frac{2}{3} = \frac{16}{3} + \frac{8}{3} = \frac{24}{3} = 8$. The value matches the table, $y = 8$.

• For $x = 11$:

Substitute into the equation:
$y = \frac{2}{3}(11) + 2\frac{2}{3} = \frac{22}{3} + \frac{8}{3} = \frac{30}{3} = 10$. The value matches the table, $y = 10$.

Thus, the equation $y = \frac{2}{3}x + 2\frac{2}{3}$ satisfies all pairs from the table, confirming it is the correct representation.

Therefore, the correct answer is $y = \frac{2}{3}x + 2\frac{2}{3}$.

To solve this problem, let's follow these steps:

• Identify key data points given in the problem.
• Determine if the data suggests a linear relationship.
• Calculate the slope using two points from the table.
• Calculate the y-intercept if necessary and match results with possible choices.

First, let's compute the slope $m$ using the first two points: $(-3, -1)$ and $(0, 0)$. The formula for the slope is:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-1)}{0 - (-3)} = \frac{1}{3}$

Next, we can check if any function $y = \frac{1}{3}x + b$ appears in the possible choices. Here since $(0, 0)$ is in the table, this suggests $b = 0$.

Thus, the equation becomes $y = \frac{1}{3}x$. This equation corresponds to choice 1. We can verify this by comparing with all given $X, Y$ pairs, and all satisfy the equation $y = \frac{1}{3} x$.

Therefore, the solution to the problem is $y = \frac{1}{3}x$.

To solve this problem, follow these steps:

• Step 1: Analyze the pattern of the given table. Notice that the increase in $Y$ for each increase of 2 in $X$ is 2, indicating a slope of 1.
• Step 2: Choose a likely linear equation from the provided options.
• Step 3: Verify which equation accurately represents all the data points.

Let's verify the equation $y = x + 1$ with all pairs:

• For $X = -3$, calculate $Y = (-3) + 1 = -2$. This matches the $Y$-value in the table.
• For $X = -1$, calculate $Y = (-1) + 1 = 0$. This too matches the $Y$-value.
• For $X = 1$, calculate $Y = 1 + 1 = 2$. Again, this matches.
• For $X = 3$, calculate $Y = 3 + 1 = 4$. This matches as well.
• For $X = 5$, calculate $Y = 5 + 1 = 6$. Finally, this matches too.

Each of the calculated $Y$-values using $y = x + 1$ aligns with the respective $Y$-values from the table.

Therefore, the correct equation that represents the function is $y = x + 1$.

To determine the equation corresponding to the values in the table, let's analyze the relationship between X and Y:

• Step 1: Check the change in Y with respect to X. The values show that each time X increases by 1, Y increases by 1, indicating a consistent change.
• Step 2: Calculate the slope $m$. Since the change is constant: $\Delta y / \Delta x = 1 / 1 = 1$. Thus, the slope $m = 1$.
• Step 3: Determine the y-intercept $b$. When X is 0, Y is -2. Thus, $b = -2$.
• Step 4: Formulate the equation.
Using the slope-intercept form, $y = mx + b$, where $m = 1$ and $b = -2$, the equation of the line is $y = x - 2$.

Verification with the table: - Substituting $X = -2$ yields $Y = -2 - 2 = -4$, which matches the table. - Similarly, substituting other values confirms consistency. Hence, this confirms our linear equation.

The corresponding equation for the function represented in the table is therefore $y = x - 2$.

To determine the correct function, we first observe the changes in $X$ and their corresponding changes in $Y$. This can help us establish the slope of the function.

Let's calculate the slope $m$ using two points from the table. From $X = -4$ to $X = 0$, $Y$ changes from -3 to -2, resulting in a change of 1 in $Y$ and 4 in $X$. Hence the slope $m = \frac{1}{4}$.

Now, let's use the point $(0, -2)$ to find the y-intercept $c$ in the equation $y = mx + c$. Substituting $m = \frac{1}{4}$ and point $(0, -2)$ in, we solve for $c$:

$-2 = \frac{1}{4}(0) + c$
$c = -2$

This gives us the equation $y = \frac{1}{4}x - 2$.

Let's verify this equation against other points in the table:

• For $X = 4$: $y = \frac{1}{4}(4) - 2 = 1 - 2 = -1$. ✓
• For $X = 8$: $y = \frac{1}{4}(8) - 2 = 2 - 2 = 0$. ✓
• For $X = 12$: $y = \frac{1}{4}(12) - 2 = 3 - 2 = 1$. ✓

All table values satisfy the equation $y = \frac{1}{4}x - 2$.

Therefore, the function that corresponds to the table is: $y = \frac{1}{4}x - 2$.

To solve this problem, we will find the linear equation that represents the function in the table by determining the slope and y-intercept.

• Step 1: Calculate the slope $m$.
  Between two points $(-3, 0)$ and $-2, 1$, the slope is:
  $m = \frac{1 - 0}{-2 - (-3)} = \frac{1}{1} = 1$
• Step 2: Use the slope and a point to find the y-intercept $b$.
  Using the point $(-3, 0)$ and the formula $y = mx + b$, plug in the values:
  $0 = 1(-3) + b$
  $0 = -3 + b$
  $b = 3$
• Step 3: Write the equation of the line:
  $y = x + 3$

This linear equation must be consistent with all the points in the table, which it is:

• $X = -2$, $Y = 1$: $1 + 3 = 4$, so correct.
• $X = -1$, $Y = 2$: $2 + 3 = 5$, so correct.
• $X = 0$, $Y = 3$: $3$, so correct.
• $X = 1$, $Y = 4$: $4 + 3 = 7$, so correct.

The calculated equation, $y = x + 3$, matches the option given as Choice 4.

Therefore, the function that corresponds to the table is $y = x + 3$.

To determine the correct equation for the function represented by the table, we will follow these steps:

• Step 1: Identify the pattern by calculating the slope $m$.

• Step 2: Determine the y-intercept $c$.

• Step 3: Select the equation that consistently matches all pairs in the table.

Step 1: Finding the slope $m$
The slope $m$ of a linear function $y = mx + c$ is given by the change in $Y$ over the change in $X$. Let's compute the slope between any two points from the table, say between $(0, -5)$ and $(1, -3)$:

$m = \frac{\Delta Y}{\Delta X} = \frac{-3 - (-5)}{1 - 0} = \frac{2}{1} = 2$

Step 2: Determining the y-intercept $c$
Using the point $(0, -5)$, we can find the y-intercept directly since $X = 0$:

$Y = 2 \times 0 + c = -5$

So, $c = -5$.

Step 3: Verifying Equation Match
The linear equation becomes $y = 2x - 5$. Substitute the remaining pairs:

- For $X = 1$, $Y = 2(1) - 5 = 2 - 5 = -3$

- For $X = 2$, $Y = 2(2) - 5 = 4 - 5 = -1$

- For $X = 3$, $Y = 2(3) - 5 = 6 - 5 = 1$

- For $X = 4$, $Y = 2(4) - 5 = 8 - 5 = 3$

All pairs check out, confirming $y = 2x - 5$ aligns perfectly.

Therefore, the solution to the problem is $y = 2x - 5$.