Representations of Functions: Matching a graph to a function

To determine the correct equation from the given choices, we observe that the graph represents a horizontal line, positioned at $y = 3$. A horizontal line is defined by a constant y-value because it does not change as x changes. Thus, the line corresponds to the equation $y = 3$, indicating this is the correct equation from the choices provided.

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

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

• Step 1: Identify the slope ($m$) of the line from the graph.
• Step 2: Determine the y-intercept ($b$) from the graph.
• Step 3: Match the slope and y-intercept to one of the given equations.

Now, let's work through each step:
Step 1: By observing the graph, we determine the slope ($m$). The line appears to pass through the points $(0, 4)$ and $(1, 5)$. Calculating the slope $m$ using the points, $m = \frac{5 - 4}{1 - 0} = 1$.

Step 2: The y-intercept $b$ is the point where the line crosses the y-axis, which is at $(0, 4)$. Therefore, $b = 4$.

Step 3: Using the slope-intercept form $y = mx + b$, substitute $m = 1$ and $b = 4$ to get $y = 1x + 4$, which simplifies to $y = x + 4$.

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

From the given choices, the correct answer is choice 4: $y = x + 4$.

To match the graph to the correct equation, we will analyze the slope and y-intercept:

• Step 1: Identify y-intercept.
  From the graph, observe that the line crosses the y-axis at $y = 4$. Therefore, the y-intercept $b = 4$.

• Step 2: Calculate slope.
  Identify two points on the graph line, such as $(0, 4)$ and $(5, 0)$. Calculate slope using $m = \frac{\Delta y}{\Delta x}$:
  $m = \frac{0 - 4}{5 - 0} = \frac{-4}{5}$.

• **Step 3: Match to equations**.
  Now we have $m = -\frac{4}{5}$ and $b = 4$, so the equation should be $y = -\frac{4}{5}x + 4$.

After comparing with the choices, we see that choice $(2)$ $y = -\frac{4}{5}x + 4$ correctly matches the information derived from the graph.

Therefore, the equation that corresponds to the graph is $y = -\frac{4}{5}x + 4$.

Let's use the below formula in order to find the slope:

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

We begin by inserting the known data from the graph into the formula:

$(0,-2),(-2,0)$

$m=\frac{-2-0}{0-(-2)}=$

$\frac{-2}{0+2}=$

$\frac{-2}{2}=-1$

We then substitute the point and slope into the line equation:

$y=mx+b$

$0=-1\times(-2)+b$

$0=2+b$

Lastly we combine the like terms:

$0+(-2)=b$

$-2=b$

Therefore, the equation will be:

$y=-x-2$

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

• Identify two clear points on the graph where the line passes through grid intersections.
• Calculate the slope of the line using these points.
• Determine the y-intercept by observing where the line crosses the y-axis.
• Compare the calculated slope and intercept with the given equations to find the match.

Let's identify two points on the line. From the graph, we see the line passes through at least two points: $(-3, 0)$ and $(0, 5)$.

Using these points, we calculate the slope $m$:

Next, observe that the y-intercept (where the line crosses the y-axis) corresponds to $y = 5$ when $x = 0$, so the y-intercept is 5.

Now we can formulate the linear equation based on our calculations:

After calculating the slope and intercept, we compare with the provided options:

• Option 1: $y = 5x$ – Slope and intercept do not match.
• Option 2: $y = -2x + 4$ – Slope and intercept do not match.
• Option 3: $y = \frac{5}{3}x + 5$ – Both slope and intercept match exactly.
• Option 4: $y = 8$ – Not a linear equation of this form.

Thus, the equation that corresponds to the function represented in the graph is:

$y = \frac{5}{3}x + 5$

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

• Step 1: Identify the type of line
  The graph illustrates a horizontal line at $y = -2$. Horizontal lines have a constant y-value across all x-values.
• Step 2: Recognize the equation of the line
  A horizontal line has the form $y = c$, where $c$ is a constant. Here, $c = -2$.
• Step 3: Match with given options
  Among the given choices, $y = -2$ correctly represents the function for the graph as it shows a line intersecting the y-axis at -2 and running parallel to the x-axis.

Therefore, the solution to the problem is $y = -2$, which corresponds to choice id "3".

To solve the problem, follow these steps:

• Step 1: Identify two clear points on the line from the graph.
• Step 2: Determine the slope using these two points.
• Step 3: Find the y-intercept using the slope-intercept form of the equation.
• Step 4: Compare the derived equation to the provided choices.

Now, let's work through each step:

Step 1: Upon examining the graph, let's assume it passes through the points $(0, 4)$ and $(3, 0)$.

Step 2: Calculate the slope $m$ using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$:
$m = \frac{0 - 4}{3 - 0} = \frac{-4}{3}$.

Step 3: Use the y-intercept point, which is already identified as $(0, 4)$. Hence, $b = 4$.

Thus, the equation of the line is $y = -\frac{4}{3}x + 4$.

Step 4: Compare this to the choices: The correct choice is $y = -\frac{4}{3}x + 4$, which matches the equation we derived.

Therefore, the solution to the problem is $y = -\frac{4}{3}x + 4$.

To solve this problem, let's examine each key feature of the graph:

• Step 1: Vertex and Direction - The graph appears to have its vertex at the origin and opens upwards.
• Step 2: Width - For a quadratic $y = ax^2$, if $|a| = 1$, the width is standard like $y = x^2$. In this graph, the parabola seems to follow the standard width.

Comparing to the options, we analyze for matching attributes:

• Choice 1: $y = 4x^2$ suggests a narrow upward parabola.
• Choice 2: $y = 2x^2$ is narrower than a standard parabola.
• Choice 3: $y = x^2$ matches a standard parabola.
• Choice 4: $y = -x^2$ opens downward.

Given the graph opens upwards and matches the standard parabola, the correct equation is $y = x^2$.

The given graph is a parabola that opens downwards and is symmetrical with respect to the y-axis, with its vertex at the origin. This is characteristic of a standard quadratic function of the form $y = -ax^2$, where $a$ is positive, indicating the parabola opens downwards because $a$ is negative in the form $y = -x^2$.

Let's compare the graph with the choices given:

• The first choice $y = x^2$ opens upwards, so it doesn't match.
• The second choice $y = -x^2$ opens downwards, matching the graph.
• The third choice $y = 3x^2$ opens upwards and is more compressed, so it doesn’t match.
• The fourth choice $y = \frac{1}{2}x^2$ opens upwards and is wider, so it doesn’t match.

Therefore, the equation that corresponds to the graph is $y = -x^2$.