Question

Which table represents a linear function?

Answer 1

Table-1 represents a linear function.

Further explanation

• A line that is not parallel to either the x-axis or the y-axis represents a line that occupies a slope or in other words a gradient.
• The gradient or steepness of a straight line is the rate at which the line rises or falls.
• The gradient is the same at any point along a straight line.
• The symbol m is used to represent the gradient or slope.

In general, the gradient of the line joining the points A(x₁, y₁) and B(x₂, y₂) is given by the formula:

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

Let's examine the slope (m) of each table.

Table-1: (1, 3), (2, 7), (3, 11), (4, 15)

(1, 3) and (2, 7) ⇒ $m = \frac{7-3}{2-1} = 4$

(2, 7) and (3, 11) ⇒ $m = \frac{11-7}{3-2} = 4$

(3, 11) and (4, 15) ⇒ $m = \frac{15-11}{4-3} = 4$

or, we choose (1, 3) and (4, 15) ⇒ $m = \frac{15-3}{4-1} = 4$

Because it has the same slope of m = 4, this table represents a linear function.

- - - - - - -

Table-2: (1, 3), (2, 8), (3, 15), (4, 21)

(1, 3) and (2, 8) ⇒ $m = \frac{8-3}{2-1} = 5$

(2, 8) and (3, 15) ⇒ $m = \frac{15-8}{3-2} = 7$

Because it does not have the same slope, this table does not represent a linear function.

- - - - - - -

Table-3: (1, 3), (2, 9), (3, 3), (4, 9)

(1, 3) and (2, 9) ⇒ $m = \frac{9-3}{2-1} = 6$

(2, 9) and (3, 3) ⇒ $m = \frac{3-9}{3-2} = -6$

Because it does not have the same slope, this table does not represent a linear function.

- - - - - - -

Table-4: (1, 3), (2, 9), (3, 27), (4, 81)

(1, 3) and (2, 9) ⇒ $m = \frac{9-3}{2-1} = 6$

(2, 9) and (3, 27) ⇒ $m = \frac{27-9}{3-2} = 18$

Because it does not have the same slope, this table does not represent a linear function.

Based on the pattern shown in Table-4, the curve represents the function of $\boxed{y = 3^x}$.

Consider the proof below:

• $x = 1 \rightarrow y = 3^1 = 3$
• $x = 2 \rightarrow y = 3^2 = 9$
• $x = 3 \rightarrow y = 3^3 = 27$
• $x = 4 \rightarrow y = 3^4 = 81$

Learn more

1. Finding the equation, in slope-intercept form, of the line that is parallel to the given line and passes through a point https://brainly.com/question/1473992
2. The midpoint https://brainly.com/question/3269852
3. Find the missing endpoint if the midpoint is known  https://brainly.com/question/5223123

Keywords: which table, represents, a linear function, gradient, slope, rate, straight, points, the formula, m = 4

Answer 2

Table 1 represents the linear function.

Further explanation:

The linear equation with slope $m$ and intercept $c$ is given as follows.

$\boxed{y = mx + c}$

The formula for slope of line with points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ can be expressed as,

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

Explanation:

In table 1,

The slope can be obtained as follows,

$\begin{aligned}m &= \frac{{7 - 3}}{{2 - 1}}\\&=\frac{4}{1}\\&= 4\\\end{aligned}$

The slope of other two points can be obtained as follows,

$\begin{aligned}m&= \frac{{11 - 7}}{{3 - 2}}\\&= \frac{4}{1}\\&= 4\\\end{aligned}$

The slopes are equal. Therefore, table 1 is correct.

In table 2,

The slope can be obtained as follows,

$\begin{aligned}m&= \frac{{8 - 3}}{{2 - 1}}\\&= \frac{5}{1}\\&=5\\\end{aligned}$

The slope of other two points can be obtained as follows,

$\begin{aligned}m&= \frac{{15 - 8}}{{3 - 2}}\\&= \frac{7}{1}\\&= 7\\\end{aligned}$

The slope sare not equal. Therefore, table 2 is not correct.

In table 3,

The slope can be obtained as follows,

$\begin{aligned}m&= \frac{{9 - 3}}{{2 - 1}}\\&= \frac{6}{1}\\&= 6\\\end{aligned}$

The slope of other two points can be obtained as follows,

$\begin{aligned}m &= \frac{{3 - 9}}{{3 - 2}}\\&=\frac{{ - 6}}{1}\\&=- 6\\\end{aligned}$

The slopes are not equal. Therefore, table 3 is not correct.

In table 4,

The slope can be obtained as follows,

$\begin{aligned}m &= \frac{{9 - 3}}{{2 - 1}}\\&= \frac{6}{1}\\&= 6\\\end{aligned}$

The slope of other two points can be obtained as follows,

$\begin{aligned}m&= \frac{{27 - 9}}{{3 - 2}}\\&= \frac{{18}}{1}\\&= 18\\\end{aligned}$

The slope is not equal. Therefore, table 4 is not correct.

Table 1 represents the linear function.

Learn more:

1. Learn more about collinear points https://brainly.com/question/5795008.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function https://brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Linear equation

Keywords: linear function, numbers, slope intercept, inequality, equation, linear inequality, shaded region, y-intercept, graph, representation.