Graphical Representation - Examples, Exercises and Solutions

The blue line is a straight line, therefore it remains constant.

Let's note that the red line is rising because it starts in the negative part (negative values) and rises to the positive part (positive values).

Therefore, the correct 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 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).

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:

$(4,2)$

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

If we plot all the points, we'll notice that point $(3,5)$ is the correct one, because:

$x=3,y=5$

And they intersect exactly on the line where the graph passes.

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'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 follow these steps:

• Step 1: Identify the key characteristics of the line parallel to $y = 4$.
• Step 2: Use the point $(1,2)$ to determine the new horizontal line equation.
• Step 3: Write the equation based on the consistent y-value of the line.

Now, let's work through each step:

Step 1: The given line $y = 4$ is a horizontal line. All horizontal lines have equations in the form $y = c$, where $c$ is a constant value describing the uniform y-position of the line.

Step 2: A line parallel to $y = 4$ that also passes through the point $(1,2)$ would maintain a constant y-value. Since it must pass through $(1,2)$, its y-intercept is $y = 2$.

Step 3: Therefore, the equation of the line parallel to $y = 4$ through $(1,2)$ is simply $y = 2$. This ensures it parallels the horizontal direction.

Thus, the algebraic representation of the line parallel to $y=4$ and passing through the point $(1,2)$ is $y = 2$.

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

• Step 1: Write the equation using point-slope form
• Step 2: Substitute the given point and slope into the equation
• Step 3: Simplify to find the slope-intercept form of the line equation

Now, let's work through each step:

Step 1: Use the point-slope form of a line equation, given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line.

Step 2: Given that the slope is represented as $2y$ and the line passes through point $(3, 7)$, we should interpret it as the slope being equivalent to 2 (as $2y$ in relation suggests $y=2x+b$ structure supposedly intended this way). This gives us a slope $m = 2$.

Using point $(3, 7)$, we substitute into the formula:

$y - 7 = 2(x - 3)$

Step 3: Simplify the equation:

$y - 7 = 2x - 6$

$y = 2x - 6 + 7$

$y = 2x + 1$

Therefore, the equation of the line is $y = 2x + 1$.

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

• Step 1: Identify the given information
• Step 2: Apply the point-slope formula
• Step 3: Convert to slope-intercept form and verify against the given choices

Now, let's work through each step:
Step 1: The problem states the line passes through point $(-5, -8)$ and has a slope of $9$.
Step 2: Using the point-slope form equation, $y-y_1 = m(x-x_1)$, plug in $(x_1, y_1) = (-5, -8)$ and $m = 9$. So the equation becomes:

Which simplifies to:

Simplifying further gives:

Then, bring the $8$ to the right side to solve for $y$ in terms of $x$:

Therefore, the equation of the line in slope-intercept form is $y = 9x + 37$, which corresponds to choice $1$.

Therefore, the solution to the problem is $y = 9x + 37$.

To solve this problem, let's proceed through these steps:

• Step 1: Identify the slope of the original line. From $y = 2x + 5$, the slope $m$ is $2$.
• Step 2: Since parallel lines have the same slope, the line we're looking for will also have a slope of $2$.
• Step 3: Use the point-slope formula with the given point $(4, 9)$:

We begin with the point-slope formula:

$y - y_1 = m(x - x_1)$

Substitute $m = 2$, $x_1 = 4$, and $y_1 = 9$ into the equation:

$y - 9 = 2(x - 4)$

Simplify the equation:

$y - 9 = 2x - 8$

Solving for $y$, we obtain:

$y = 2x - 8 + 9$

$y = 2x + 1$

Therefore, the algebraic representation of the line parallel to $y = 2x + 5$ that passes through $(4, 9)$ is:

$y = 2x + 1$

To solve this problem, we begin by noting that since the line is parallel to $y = 3x + 4$, it must have the same slope, $m = 3$.

We use the point-slope form of the equation of a line, which is:

Here, the slope $m = 3$ and the line passes through the point $\left(\frac{1}{2}, 1\right)$. Therefore, we substitute these values into the point-slope formula:

Next, we simplify this equation:

• Distribute the slope $3$ on the right side:
$y - 1 = 3x - \frac{3}{2}$
• Add 1 to both sides to solve for $y$:
$y = 3x - \frac{3}{2} + 1$
• Simplify $-\frac{3}{2} + 1$:
$y = 3x - \frac{1}{2}$

Thus, the equation of the line parallel to $y = 3x + 4$ and passing through the point $\left(\frac{1}{2}, 1\right)$ is:

The corresponding choice is:

$y=3x-\frac{1}{2}$

To determine the line's equation, we'll follow these steps:

• Use the point-slope form of a line, given by $y - y_1 = m(x - x_1)$.
• Substitute $m = \frac{1}{2}$, $x_1 = 5$, and $y_1 = 17\frac{1}{2}$ into the equation.
• Solve for $y$ to put the equation in slope-intercept form.

Now, let's work through the steps:

Given the point $(5, 17\frac{1}{2})$ and slope $m = \frac{1}{2}$, our start point is the point-slope form:
$y - 17\frac{1}{2} = \frac{1}{2}(x - 5)$.

Convert the mixed number to an improper fraction: $17\frac{1}{2} = \frac{35}{2}$.

Thus, the equation becomes $y - \frac{35}{2} = \frac{1}{2}(x - 5)$.

Distribute the slope on the right-hand side:
$y - \frac{35}{2} = \frac{1}{2}x - \frac{5}{2}$.

To solve for $y$, add $\frac{35}{2}$ to both sides:
$y = \frac{1}{2}x - \frac{5}{2} + \frac{35}{2}$.

Combine the fractions on the right-hand side:
$y = \frac{1}{2}x + \frac{30}{2}$, which simplifies to $y = \frac{1}{2}x + 15$.

Therefore, the equation of the line in slope-intercept form is $y = \frac{1}{2}x + 15$.

Comparing this with the multiple-choice options, the correct answer is:

$y = \frac{1}{2}x + 15$