Graphical Representation: Finding points on the graph of a function

To find which point lies on the graph of the function $y = 2x + 1$, we will substitute the $x$ values from each choice and see if the output $y$ matches the $y$ of that point.

Let's evaluate each point:

• For $(0,0)$: Substitute $x = 0$ into $y = 2x + 1$, we get $y = 2(0) + 1 = 1$. Does not match $y = 0$.
• For $(3,10)$: Substitute $x = 3$ into $y = 2x + 1$, we get $y = 2(3) + 1 = 7$. Does not match $y = 10$.
• For $(5,12)$: Substitute $x = 5$ into $y = 2x + 1$, we get $y = 2(5) + 1 = 11$. Does not match $y = 12$.
• For $(4,9)$: Substitute $x = 4$ into $y = 2x + 1$, we get $y = 2(4) + 1 = 9$. Matches $y = 9$.

Therefore, the point through which the function $y = 2x + 1$ passes is $(4,9)$.

To solve this problem, we need to check each given point to determine if it lies on the line represented by the equation $y = \frac{1}{2}x + \frac{3}{4}$.

• Check Point (1): $(1, \frac{4}{5})$
  Substitute $x = 1$ into the function:
  $y = \frac{1}{2}(1) + \frac{3}{4} = \frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4} \neq \frac{4}{5}$.
  The point does not lie on the line.

• Check Point (2): $(1, 2\frac{1}{4})$
  Substitute $x = 1$:
  $y = \frac{1}{2}(1) + \frac{3}{4} = \frac{5}{4} \neq 2\frac{1}{4}$.
  The point does not lie on the line.

• Check Point (3): $(3, 2\frac{1}{4})$
  Substitute $x = 3$:
  $y = \frac{1}{2}(3) + \frac{3}{4} = \frac{3}{2} + \frac{3}{4} = \frac{6}{4} + \frac{3}{4} = \frac{9}{4} = 2\frac{1}{4}$.
  The point lies on the line.

• Check Point (4): $(4, 2\frac{3}{4})$
  Substitute $x = 4$:
  $y = \frac{1}{2}(4) + \frac{3}{4} = 2 + \frac{3}{4} = \frac{8}{4} + \frac{3}{4} = \frac{11}{4} = 2\frac{3}{4}$.
  The point lies on the line.

The points $(3, 2\frac{1}{4})$ and $(4, 2\frac{3}{4})$ satisfy the equation, indicating that these points are on the line.
Therefore, the solution is Answers C and D are correct.

To determine through which points the function $y = 2(x+2) - 1$ passes, let's proceed methodically.

Step 1: Simplify the expression of the linear function:

The given function is $y = 2(x+2) - 1$. Distribute the $2$ over the terms inside the parenthesis:
$y = 2x + 4 - 1$

Simplify further by combining like terms:
$y = 2x + 3$

Step 2: Evaluate the function at different $x$-values to find which points lie on the line represented by the function $y = 2x + 3$.

Step 3: Check each of the provided choice points:

• For choice $(0,0)$: Substituting $x = 0$ into $y = 2x + 3$, we get $y = 2(0) + 3 = 3$. This does not match $y = 0$.
• For choice $(10,13)$: Substituting $x = 10$ into $y = 2x + 3$, we get $y = 2(10) + 3 = 23$. This does not match $y = 13$.
• For choice $(5,13)$: Substituting $x = 5$ into $y = 2x + 3$, we get $y = 2(5) + 3 = 13$. This matches $y = 13$, so this point lies on the line.
• For choice $(4,13)$: Substituting $x = 4$ into $y = 2x + 3$, we get $y = 2(4) + 3 = 11$. This does not match $y = 13$.

Therefore, the point through which the given function passes is $(5,13)$.

To solve this problem, we'll verify whether the point $(4, 19)$ lies on the graph of the function $y = 2(x + 6) - 1$.

• Substitute $x = 4$ into the function.
• Calculate the resulting y-value.
• Compare this y-value to 19 to determine if $(4, 19)$ is on the graph.

Let's perform these steps:

Step 1: Substitute $x = 4$ into the function:

$y = 2(4 + 6) - 1$

Step 2: Simplify the expression:

$y = 2 \times 10 - 1 = 20 - 1 = 19$

The calculated y-value is 19.

Step 3: Compare the calculated y-value with the y-coordinate of the point:

Since the calculated y-value (19) matches the y-coordinate of the point (19), the point $(4, 19)$ does indeed lie on the graph.

Thus, the answer is: Yes (the point lies on the graph).

To solve this problem, we'll proceed by comparing each answer choice through substitution:

• Choice 1: $y = -4x$
  Substitute $x = \frac{1}{2}$: $y=-4\times\frac{1}{2}=-2$
  Does this equal 5? No, so this choice is incorrect.

• Choice 2: $y = 4x$
  Substitute $x = \frac{1}{2}$: $y=4\times\frac{1}{2}=2$
  Does this equal 5? No, so this choice is also incorrect.

• Choice 3: $y = -4x + 3$
  Substitute $x = \frac{1}{2}$: $y=-4\times\frac{1}{2}+3=-2+3=1$
  Does this equal 5? No, this choice is incorrect as well.

• Choice 4: $4x = y - 3$
  Substitute $x = \frac{1}{2}$ and $y = 5$: $4 \times \frac{1}{2} = 5 - 3 \$ which simplifies to$2 = 2 \$
  This equation holds true with the given point.

Therefore, the solution to the problem is $4x = y - 3$.

To solve this problem, we will check which of the given functions passes through the point $(2, 8)$ by substituting $x = 2$ and verifying if the resulting $y$-value is 8.

• Check each function:
• 1. For $y = \frac{x}{4}$:

Substitute $x = 2$:
$y = \frac{2}{4} = \frac{1}{2}$
This does not satisfy the condition $y = 8$.

• 2. For $y = 4x$:

Substitute $x = 2$:
$y = 4 \times 2 = 8$
This satisfies the condition $y = 8$.

• 3. For $y = -4x$:

Substitute $x = 2$:
$y = -4 \times 2 = -8$
This does not satisfy the condition $y = 8$.

• 4. For $y = x^4$:

Substitute $x = 2$:
$y = 2^4 = 16$
This does not satisfy the condition $y = 8$.

Therefore, the only function that passes through the point $(2, 8)$ is $y = 4x$.

Thus, the solution to the problem is $y = 4x$.

To determine which point the function $y = \frac{3}{4}x + 4$ passes through, we will evaluate the given choices:

• Evaluate the function for $x = \frac{1}{2}$:

$y = \frac{3}{4} \times \frac{1}{2} + 4$

$y = \frac{3}{8} + 4$

$y = \frac{3}{8} + \frac{32}{8}$

$y = \frac{35}{8}$

Thus, the coordinate $\left(\frac{1}{2}, \frac{35}{8}\right)$ lies on the graph.

Therefore, the function passes through the point $\left(\frac{1}{2}, \frac{35}{8}\right)$.

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

• Step 1: Identify the given point and convert to uniform representation.
• Step 2: Substitute the point into each equation to find which is satisfied.

Now, let's work through each step:
Step 1: The problem provides the point $\left(\frac{1}{2}, 4\frac{1}{2}\right)$, which can be expressed as $\left(\frac{1}{2}, \frac{9}{2}\right)$.
Step 2: We will substitute $x = \frac{1}{2}$ and $y = \frac{9}{2}$ into each equation:

Choice 1: $2 - y = 5x$

Substitute: $2 - \frac{9}{2} = 5 \times \frac{1}{2}$
Simplify: $-\frac{5}{2} = \frac{5}{2}$. This equation is not satisfied.

Choice 2: $5x = y - 2$

Substitute: $5 \times \frac{1}{2} = \frac{9}{2} - 2$
Simplify: $\frac{5}{2} = \frac{9}{2} - \frac{4}{2} = \frac{5}{2}$. This equation is satisfied.

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

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

• Step 1: Simplify the expression for $y = 6(2x+4) + x$.
• Step 2: Calculate $y$ at $x = 0$.

First, let's simplify the expression for $y$:

$y = 6(2x + 4) + x$
$= 6 \times 2x + 6 \times 4 + x$
$= 12x + 24 + x$
$= 13x + 24$

Now, let's evaluate $y$ when $x = 0$:

$y = 13(0) + 24$
$= 0 + 24$
$= 24$

This means the function passes through the point $(0, 24)$. Therefore, the solution to the problem is $(0, 24)$.

We want to find which function passes through the point $(6, 80)$. This means we need the function to satisfy the equation $y = 80$ when $x = 6$.

Let's evaluate the candidate functions one by one:

• Function 1: $y = 5(x + 10)$
  Substitute $x = 6$: $y = 5(6 + 10) = 5 \times 16 = 80$.
  This function gives the correct output, so it passes through the point $(6, 80)$.

• Function 2: $y = x$
  Substitute $x = 6$: $y = 6$.
  The output $6$ is not equal to $80$, so this function does not pass through $(6, 80)$.

• Function 3: $y = 2x + 3$
  Substitute $x = 6$: $y = 2(6) + 3 = 12 + 3 = 15$.
  The output $15$ is not equal to $80$, hence this function does not pass through $(6, 80)$.

• Function 4: $y = 12 - 5x$
  Substitute $x = 6$: $y = 12 - 5(6) = 12 - 30 = -18$.
  The output $-18$ is not equal to $80$, hence this function does not pass through $(6, 80)$.

After checking all options, the function $y = 5(x + 10)$ is the only one that satisfies the condition. Therefore, the correct answer is $y = 5(x + 10)$.

To solve the problem, let's analyze each equation to determine which function's graph passes through the point $(5, 25)$.

• Equation 1: $y = 4x + 1$
  Substitute $x = 5$ into equation: $y = 4(5) + 1 = 20 + 1 = 21$
  This does not satisfy $y = 25$.
• Equation 2: $y = 5x + x$
  Substitute $x = 5$ into equation: $y = 5(5) + 5 = 25 + 5 = 30$
  This does not satisfy $y = 25$.
• Equation 3: $y = 4x + x$
  Substitute $x = 5$ into equation: $y = 4(5) + 5 = 20 + 5 = 25$
  This satisfies $y = 25$.
• Equation 4: $y = 3x + x$
  Substitute $x = 5$ into equation: $y = 3(5) + 5 = 15 + 5 = 20$
  This does not satisfy $y = 25$.

Upon evaluating all the options, Equation 3, $y = 4x + x$, is the only one that satisfies the condition $y = 25$ when $x = 5$. Therefore, the function whose graph passes through the point $(5, 25)$ is $y = 4x + x$.

To determine through which point the function passes, we begin by simplifying the given equation.

Given: $x = y - 4 + 2x$

Rearranging the terms to solve for $y$:

$x = y - 4 + 2x$

Subtract $x$ from both sides to isolate the terms involving $y$:

$0 = y - 4 + x$

Rearrange to solve for $y$:

$y = -x + 4$

Now, we will test each point to see which satisfies the equation $y = -x + 4$.

• For $(-1, 5)$, substitute $x = -1$:

$y = -(-1) + 4 = 1 + 4 = 5$

This point satisfies the equation.

• For $(0, 5)$, substitute $x = 0$:

$y = -(0) + 4 = 4$

This point does not satisfy the equation.

• For $(1, 5)$, substitute $x = 1$:

$y = -(1) + 4 = -1 + 4 = 3$

This point does not satisfy the equation.

• For $(2, 5)$, substitute $x = 2$:

$y = -(2) + 4 = -2 + 4 = 2$

This point does not satisfy the equation.

Therefore, the graph of the function passes through the point $(-1, 5)$.