Parabola of the Form y=x²+c: Linking function properties to its representation

To solve this problem, we will determine the vertical shift given to the parent function $y = x^2$ to form the observed parabola.

• Identify that the problem involves a vertical translation of the parabola $y = x^2$.

• The function takes the form $y = x^2 + c$, where $c$ indicates the vertical shift.

• From the graph given, it is seen that the vertex of the parabola is situated at $y = -6$ when viewed from the intersection with the y-axis.

• This downward shift corresponds to the constant $c$ being negative, specifically $c = -6$.

• By this observation, the function becomes $y = x^2 - 6$.

Therefore, the solution to the problem is $y = x^2 - 6$, matching choice 2.

This problem involves determining the algebraic representation of a parabola that was presented graphically. Our goal is to interpret the graph and express it in terms of its equation for a downward-opening parabola.

To solve the problem, follow these steps:

• Step 1: Identify the Parabola's Vertex – According to the diagram, the vertex is positioned at $(0, 1)$, implying that at $x = 0$, the maximum value of $y$ is 1. This indicates that the constant term $c$ in the parabola's equation will be 1.
• Step 2: Determine the Parabola’s Orientation – The given parabola is described as downward-opening. This means the coefficient in front of $x^2$ must be negative. This leads us to the formula $y = -x^2 + c$.
• Step 3: Construct the Equation – With the downward orientation and the vertex point established, the equation becomes $y = -x^2 + 1$ as $c = 1$ from the vertex.

By matching one of the multiple-choice answers with our derived equation, it's clear that choice 2 corresponds to $y = -x^2 + 1$. Thus

Therefore, the algebraic representation of the function is $y = -x^2 + 1$.

In this problem, we are tasked with identifying the algebraic representation of a function given a graphical depiction. Given the problem's indication that we are dealing with parabolas, particularly those of the form $y = x^2 + c$, we need to examine the provided graph for features typical of this family of functions.

The graph structure in the problem suggests a parabolic curve, centered symmetrically, which is indicative of the simplest unmodified parabola, $y = x^2$. The vertex likely lies at the origin, and the parabola opens upwards, a key characteristic of the function $y = x^2$ when the coefficient of $x^2$ is positive and equal to 1.

Upon reviewing the multiple-choice options, the expression that corresponds to this graph is:

• Option 1: $y = x^2$

Therefore, the algebraic representation that corresponds to the function is $y = x^2$.

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

• Identify the graphical points where the parabola interacts with the axes.
• Determine the role of any numbers indicated along the axes, specifically for vertical shifts.
• Write the function based on these observations.

Now, let's work through each step:
Step 1: The parabola is shown as an intersection with a horizontal line labeled "4". This suggests that the parabola is shifted upwards by 4 units.
Step 2: The standard quadratic equation without shifts is $y = x^2$. Therefore, to account for a shift of 4 units upwards, we modify this to $y = x^2 + 4$.
Step 3: With this shift identified, the algebraic representation of the function is completed.

Therefore, the solution to the problem is $y = x^2 + 4$.

The problem involves determining the algebraic formula of a function represented by a graph of a parabola. From the axes on the graph, we observe that the graph intersects the y-axis at $y = -2$. This intersection indicates that the constant term $c$ in the quadratic function $y = x^2 + c$ is $-2$.

• Step 1: Start with the parabola formula $y = x^2 + c$. Since this is a standard vertical parabola centered on the y-axis (no $x$ term with a coefficient), it takes the form $y = ax^2 + c$.
• Step 2: From the graph, we see that when $x = 0$, $y = -2$. Substituting into the equation gives us: $y = 0^2 + c = -2$.
• Step 3: Solve for $c$. Here, it is evident that $c = -2$ since the equation simplifies directly to this when $x = 0$.

Therefore, the equation of the parabola is $y = x^2 - 2$, which corresponds to choice 3 in the provided options.

Therefore, the solution to the problem is $y = x^2 - 2$.

To solve this problem, we need to identify the algebraic representation of the given graph of a function. Based on the observation of the parabola, which opens downward and intersects the y-axis at the point $y = 6$, we can deduce its form.

Step 1: Identify the type of parabola.
The graph shows a parabola opening downwards, indicating that the leading coefficient $a$ in the equation $y = ax^2 + c$ is negative.

Step 2: Identify key points on the graph.
Since the parabola intersects the y-axis at $y = 6$, this means when $x = 0$, $y = 6$. Thus, the equation $y = ax^2 + c$ simplifies to $y = -x^2 + 6$.

Step 3: Confirmation of the algebraic representation.
From the downward orientation and the vertical intersection at $y = 6$ without any lateral shifts, the equation $y = -x^2 + 6$ fully describes the parabola.

Therefore, the corresponding algebraic representation of the function is $y = -x^2 + 6$.

We begin by analyzing the shape and nature of the parabola described. From the visual description, the parabola opens downwards, indicating that the leading coefficient of the quadratic must be negative.

Notice that the vertex of the parabola sits on the negative direction of the y-axis, which is consistent with the vertex form $y = ax^2 + k$. For a parabola opening downwards, we have $a < 0$.

Given that the vertex appears at the value $y = -2$ on the y-axis, we can leverage the standard form:

The function base form becomes $y = -x^2 + c$.

The detail given suggests a vertex directly on $y = -2$ (as one of the intersecting point/vertex specifics), hence: $y = -x^2 - 2$.

The mathematical representation of this function, aligned with the vertex downwards and y-intercept at -2, is therefore $y = -x^2 - 2$.

To solve this problem, we need to identify how the given parabola is translated from its standard form.

The standard form of a parabola is $y = x^2$. When a vertical translation occurs, the equation becomes $y = x^2 + c$, where $c$ shifts the parabola up or down along the y-axis.

In the graph, there is an indication that the minimum point (or vertex) of the parabola has been shifted upwards so that it crosses the $y$-axis at the point where $y = 5$. This tells us that the entire parabola has been shifted vertically upwards by 5 units. Therefore, $c = 5$.

Thus, the algebraic representation of the translated function is:

$y = x^2 + 5$

This matches exactly with choice 3 from the provided options.

Therefore, the solution to the problem is $y = x^2 + 5$.