Graphical Representation: Identifying the algebraic expression of a linear function

Let's first remember that the origin of the coordinate system is $(0,0)$.

We'll highlight the point on the graph, noting that it doesn't lie on any of the plotted lines.

Therefore, the answer is C; If we plot the point $(3,1)$, then we'll see that it lies on line I (the blue one).

Let's pay attention to the point where the lines intersect. We'll mark it.

We'll find that:

$X=4,Y=2$

Therefore, the point is:

$(4,2)$

Note that the line intersects only the Y-axis. In other words, it does not go through the X-axis at all.

Therefore, the answer is (d).

To solve this problem, we need to identify which graph fulfills two criteria: intersecting the origin and having a positive slope (i.e., being an increasing function).

Let's examine the provided graphs:

• Criterion 1: Intersects the Origin
  A graph that intersects the origin will pass through the point $(0,0)$. This means that when $x = 0$, $y$ should also be $0$.

• Criterion 2: Increasing Function
  An increasing function is indicated by a line that has a positive slope. This means that as $x$ increases, $y$ should also increase.

Analysis of Graphs:

• The Green Graph: This graph passes through the point (0,0) but moves from the top left to the bottom right, which represents a negative slope.

• The Blue Graph: This graph also does not pass through the origin; it intersects the y-axis above the origin point.

• The Yellow Graph: This graph intersects below the origin and slants negatively, indicating a negative slope.

• The Red Graph: This graph passes through the point (0,0) and moves from the bottom left to the top right, which confirms a positive slope. Therefore, it is an increasing function that intersects the origin.

Based on the analysis above, the graph that represents an increasing function that intersects the origin is confidently identified as the red graph.

Therefore, the correct choice is $\text{the red graph}$.

Note that in answer A there is an exponent, therefore the answer is incorrect.

Note that in answer C, if we multiply X by X we get X to a power, therefore the answer is incorrect.

Note that in answer D there is an exponent, therefore the answer is incorrect.

In answer B the following formula can be observed.

$y=mx+b$

To solve this problem, we need to determine which representations describe a linear function by analyzing each given choice:

• Choice 1: $y = 1 - 2x$
  - This is in the form $y = mx + b$, where $m = -2$ and $b = 1$, making it a linear function.
• Choice 2: $y = -2x^2 + x$
  - The term $x^2$ indicates a quadratic polynomial, which is not linear due to the power of 2 on $x$.
• Choice 3: $y = x$
  - In the form $y = mx + b$, it is $y = 1x + 0$, with $m = 1$ and $b = 0$, thus a linear function.
• Choice 4: Asserts that both Choice 1 and Choice 3 are correct, which aligns with our analysis.

Based on this examination, choices forming linear functions are ones where the equation stays in the standard linear form $y = mx + b$ with no additional exponents or variable products. Thus, the correct answer is:

Answers A + C are correct

To solve this problem, we'll analyze each pair of given equations to see if they are linear and parallel.

Let's examine each pair:

• Choice 1:
  $y = \frac{1}{2}x + 10$
  $y = \frac{1}{2}(x + 2)$ simplifies to $y = \frac{1}{2}x + 1$
  Both equations are linear with the same slope of $\frac{1}{2}$, indicating they are parallel.
• Choice 2:
  $y = 3(x + 4)$ simplifies to $y = 3x + 12$
  $y = 3x^2 + 12$ is not in the form $y = mx + b$ as it includes an $x^2$ term. Thus, it is non-linear.
• Choice 3:
  $y = 5 + 12x$ is already in the form $y = mx + b$ with $m = 12$
  $y = 5 + 12 + x$ simplifies to $y = x + 17$, which has a slope of 1.
  Slopes are different, so not parallel.
• Choice 4:
  $y = 3x + 2$, slope $m = 3$
  $y = 2x + 3$, slope $m = 2$
  Different slopes, thus not parallel.

Therefore, based on our analysis, the correct choice is Choice 1:

$y = \frac{1}{2}x + 10$ and $y = \frac{1}{2}(x + 2)$

To solve this problem, we'll examine each expression to see if it represents a linear function:

• Option A: $y = 5 - 3x$.
  This expression fits the linear form $y = mx + c$ with $m = -3$ and $c = 5$. Hence, it is a linear function.
• Option B: $y = -4(x+1) + 4x$.
  First, expand the expression:
  $-4(x+1) = -4x - 4$.
  Substituting, we get $y = -4x - 4 + 4x$, which simplifies to $y = -4$.
  This is a linear function where $y = c$ (a constant term with zero slope).
• Option C: $y = -3x^2 + 2$.
  The term $-3x^2$ indicates a quadratic expression, as the highest power of $x$ is 2. Therefore, it is not a linear function.
• Option D: $y = 6 + x^3$.
  The term $x^3$ shows a cubic expression since the highest power of $x$ is 3, thus not a linear function.

Clearly, only options A and B describe linear functions. Therefore, the correct answer is:

Answers A and B are correct.

To solve this problem, we'll examine each expression to verify which represents a linear function:

• Option A: $x = y - 4$

This is a linear function. It can be written in the form $y = x + 4$, which matches the linear form $y = mx + c$ with $m = 1$ and $c = 4$.

• Option B: $x = 3x^2 + 1$

This equation involves a squared term ($x^2$), which means it's not a linear function. Linear functions do not have variables raised to powers other than one.

• Option C: $x = x + y - 1$

Rearrange to isolate $y$:

$y = 1$. This is a linear equation representing a horizontal line in the xy-plane.

• Option D: $x = x^2 + 4 - y$

This equation also involves a squared term ($x^2$), which disqualifies it as a linear function.

Based on this analysis, both Options A and C describe linear functions, and therefore the correct answer is that Answers A and C are correct.

To solve this problem, we'll examine each given choice:

• Choice 1: $y = -x$ and $y = x$. Both can be written in the form $y = mx + b$. Slopes are $-1$ and $1$, hence not parallel.

• Choice 2: $y = 1 + x^2$ and $y = 2 + x^2$. These are quadratic forms, not linear equations.

• Choice 3: $y = 2(x+1)-x$ simplifies to $y = (2-x) + 2$, which further reduces to $y = x + 2$, hence $y = x + 2$.
  - Both equations, $y = x + 2$ and $y = x + 2$, are in linear form with equal slopes of $1$. They are the same line, hence parallel by default.

• Choice 4: $y = 2 + x$ is the same as $y = x + 2$. - $y = x$ compares with $y = x + 0$.
  - Slopes of both are $1$, hence they are parallel.

• Choice 5: Claims C and D are correct, which entails verifying that both choices depict linear functions and parallel lines as previously identified.

Upon analysis, choices C and D both represent linear functions and their line pairs have equal slopes, indicating parallel lines. Thus, the correct answer is that both choices C and D are correct.

Therefore, the correct answer to the problem is: Choices C and D are correct.

To solve this problem, we'll examine each pair of equations to determine which consists of linear functions with parallel lines.

• Step 1: Identify the form of the equation for each choice and ensure they are linear if they can be written as $y = mx + b$.

• Step 2: Calculate or identify the slope for each equation to compare within the pair.

Now, consider each given choice:

Choice 1:
- Equation 1: $y = 5x$ has a slope of 5.
- Equation 2: $y = 5(x^2 + 1)$ simplifies to a nonlinear form because of the $x^2$ term, so it is not relevant for parallelism in linear functions.

Choice 2:
- Equation 1: $y = x$ has a slope of 1. - Equation 2: $y = -x + x + 10$ simplifies to $y = 10$ which is a constant and does not form a linear equation with variable terms, thus irrelevant.

Choice 3:
- Equation 1: $y = 2(x + 3)$ simplifies to $y = 2x + 6$, slope is 2.
- Equation 2: $y = -2(3 + x)$, simplifies to $y = -6 - 2x$, slope is -2.
- Slopes are not equal, lines are not parallel.

Choice 4:
- Equation 1: $y = -4(x + 1)$ simplifies to $y = -4x - 4$, slope is -4.
- Equation 2: $y = 8x - 12(x + 1)$ simplifies to:
$y = 8x - 12x - 12$, which simplifies to $y = -4x - 12$, slope is -4.
Both slopes are -4, indicating these are parallel lines.

Therefore, the correct choice is Choice 4:

_^$y=-4(x+1)$

_{^{$y=8x-12(x+1)$}}