Equations with Absolute Values: Identifying functions from graphs

To solve this problem, we'll conduct the following steps:

• Step 1: Identify the vertex of the graph shown.
• Step 2: Compare the vertex to each function given in the choices.
• Step 3: Select the function whose vertex matches the graph.

Now, let’s work through these steps:

Step 1: The graph shows a vertex at the point (5, 0). This suggests that the graph is the absolute value function centered at $x = 5$.

Step 2: We compare this with each available function:
- $|x-5|$: This corresponds to a vertex at $(5, 0)$.
- $|x-3|$ would give a vertex at $(3, 0)$.
- $|x|$ would give a vertex at $(0, 0)$.
- $|5x|$ involves a similar transformation but with varied scaling and zero vertex.

Step 3: Therefore, the function $|x-5|$ perfectly matches the vertex (5, 0) of the graph plotted.

Our analysis confirms that the absolute value function corresponding to the graph is $|x-5|$.

To solve this problem, we'll analyze the graph to match it with the provided equations.

• Step 1: The graph is in a 'V' shape presenting symmetry around the y-axis and has its vertex at the origin.
• Step 2: The equation that corresponds to this type of graph is the absolute value function $y = |x|$. Absolute value functions show symmetry around the y-axis and have a vertex where the argument of the absolute value, here $x$, equals zero.
• Step 3: The graph we see perfectly matches an absolute value function, as it does not extend below the x-axis and has the aforementioned symmetrical properties. Therefore, it is represented by the equation $y = |x|$.

Therefore, the solution to the problem is $y = |x|$, matching the graph with the choice corresponding to $y = |x|$.

Let's proceed to identify the correct function corresponding to the graph:

The graph displays a "V" shape, which is a strong indicator of an absolute value function. To identify it, we must find its vertex.

Given the transformation properties of absolute values, the graph equation is of the general form $y = \lvert x - h \rvert + k$, where $(h, k)$ is the vertex.

• The vertex for the graph is found at $(1, 0)$.
• This means the function template follows $y = \lvert x - 1 \rvert$. This aligns with an $x$ translation of 1 to the right.

Checking the answer choices given:

• Choice (1): $y = x - 1$ - This represents a straight line, not an absolute value function.
• Choice (2): $y = x + 1$ - Incorrect due to the vertex position.
• Choice (3): $y = |1x|$ - Incorrect due to no translation present: the vertex is unaffected by modifying the coefficient of $x$.
• Choice (4): $y = \lvert x - 1 \rvert$ - Matches the $(1, 0)$ vertex position and absolute value function type.

Thus, the correct answer is $y = \lvert x - 1 \rvert$.

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

• Step 1: Analyze the graph provided. The graph appears to have a distinct 'V' shape, indicating that at least part of the graph is reflected about the x-axis.
• Step 2: Recognize that a standard quadratic function $f(x) = x^2 - 3$ would have a vertex at (0, -3) if unaltered. Any sections of this below the x-axis being reflected upwards suggests the graph of an absolute value function.
• Step 3: Apply the absolute value function: When the function is $f(x) = |x^2 - 3|$, all values of $f(x)$ which are below the x-axis due to $x^2$ terms less than 3 are reflected above the x-axis.

We can eliminate any function that does not involve an absolute value or shift consistent with reflected x-intercepts. Comparing with each choice:

• Choice 1 $f(x) = x^2 - 3$ is incorrect because the graph wouldn't show the V-like reflection above the x-axis.
• Choice 2 $f(x) = |x^2 - 3|$ matches perfectly since this transformation reflects all negative portions of the $x^2 - 3$ graph above the x-axis as observed.
• Choice 3 $f(x) = |x^2 + 3|$ is incorrect as this has no crossing or vertex shifts indicating a reflection starting point below the x-axis.
• Choice 4 $f(x) = x^2$ doesn't reflect along y-values shifted down from 3.

Therefore, the function that corresponds to the graph is $f(x) = |x^2 - 3|$.

We will determine the functions represented by the provided graphs. There are two distinct graphs: a V-shaped graph and a horizontal line. Let's identify each:

1. Observing the V-shaped blue graph, we interpret it as an absolute value function. The shape of such graphs is $f(x) = |x-a|$, where the vertex occurs at $x = a$. The vertex sits at the lowest/highest point, most likely along the line $x = 5$. This suggests $f(x) = |x-5|$.

2. The horizontal red line is a constant function where $g(x) = c$ is consistent across all values of $x$. This line crosses at $y = 2$, hence, $g(x) = 2$.

Thus, the functions are:

• $f(x) = |x-5|$
• $g(x) = 2$

Upon evaluation, these equations best fit the graph representations.

The solution to the problem is $f(x) = |x-5|$ and $g(x) = 2$.

The problem involves determining the equations corresponding to a blue absolute value graph and a red horizontal line based on a given graph.

Firstly, consider the blue absolute value graph:

• It is symmetric and has a "V" shape, typically expressed as $f(x) = |x - h| + k$.
• The vertex, or minimum point, of this graph is at (6, 6). Hence, the equation can be written as $f(x) = |x - 6|$, because at $x = 6$, the function value is at its minimum.
• The value of $k$, the vertical shift, is not explicitly needed here since the vertex already indicates the graph reaches down to the x-axis and $f(x) = |x - 6| + 0$.

Next, assess the red horizontal line:

• This line crosses the y-axis at $y = 3$, which aligns with the horizontal segment at points (3, 3) and (9, 3).
• Thus, the constant function equation is $g(x) = 3$.

Upon evaluation of the given choices, choice 1 corresponds correctly with these interpretations:

$\begin{cases} f(x)= |x-6| \\ g(x)=3 \end{cases}$

Therefore, the correct set of functions is $f(x) = |x - 6|$ and $g(x) = 3$.

First, we identify the absolute value function associated with the V-shaped graph. The vertex of this V-shape is at the point $(-6, 0)$, indicating an expression of $f(x) = |x + 6|$.

Next, observe the red horizontal line, which consistently passes through $y = 3$. This confirms that the horizontal line function is $g(x) = 3$.

Having identified both functions, we conclude that they correspond to:

$\begin{cases} f(x)= |x+6| \\ g(x)=3 \end{cases}$