Parabola of the form y=x²: Determine whether the function is defined at a certain point

The problem asks us to find an $x$ such that in the function $y = x^2$, the value of $y$ becomes 16. To do this, we'll substitute $y = 16$ into the equation and solve for $x$.

1. Start with the equation of the function:

$y = x^2$

2. Substitute $y = 16$ into the equation:

$16 = x^2$

3. Solve $x^2 = 16$ for $x$:

• Take the square root of both sides to solve for $x$:
• $x = \pm \sqrt{16}$
• This gives $x = 4$ or $x = -4$

4. Identify the points on the function for these values of $x$:

• For $x = 4$, the point is $(4, 16)$.
• For $x = -4$, the point is $(-4, 16)$, but this is not provided in the choice list.

Among the given options, the point we find in the choices is:

$(4, 16)$

Therefore, the correct answer is the choice that corresponds with this point:

$(4,16)$

To determine if there is a point on the graph of the parabola $y = x^2$ where $y = 4$, we need to find values of $x$ that satisfy the equation $x^2 = 4$.

Let's solve the equation step by step:

• Set the equation: $x^2 = 4$.
• Take the square root of both sides to solve for $x$:
• $x = \sqrt{4}$ or $x = -\sqrt{4}$.
• This gives us $x = 2$ or $x = -2$.

Therefore, the points on the graph where $y = 4$ are $(2, 4)$ and $(-2, 4)$.

This matches the provided correct answer of $(2, 4)$ and $(-2, 4)$.

Therefore, the correct solution is the point set $(2, 4)$ and $(-2, 4)$.

To determine if the function $y = x^2$ passes through the point $(6, 36)$, follow these steps:

• Step 1: Identify the given point and function. We have $x = 6$ and we need to check if $y = 36$ when $y = x^2$.
• Step 2: Substitute $x = 6$ in the function $y = x^2$:
  $y = 6^2 = 36$.
• Step 3: Compare the calculated $y$ value (36) to the given value (36).

Since the calculated value of $y$ is equal to the given value, the function $y = x^2$ indeed passes through the point $(6, 36)$.

Therefore, the answer is Yes.

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

• Step 1: Identify the given function and required y-value.
• Step 2: Attempt to solve $x^2 = -6$ for $x$.
• Step 3: Conclude based on the results of the equation.

Let's work through each step:

Step 1: The function we are dealing with is $y = x^2$, and we need to find $x$ such that $y = -6$.

Step 2: Substitute $-6$ for $y$, which gives us:

$x^2 = -6$

Step 3: To solve $x^2 = -6$, consider whether it is possible for a real number squared to equal a negative number. A critical point here is that the square of any real number is non-negative. Therefore, there is no real value of $x$ that satisfies $x^2 = -6$.

Thus, there is no point where $y = -6$ for the function $y = x^2$.

The correct answer is No.

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

• Step 1: Analyze the equation $y = x^2$.
• Step 2: Investigate whether a negative $y$-value is possible.

Now, let's work through each step:
Step 1: The function we have is $y = x^2$. This function is defined for all real numbers and always gives a non-negative value $y$ because squaring a real number cannot result in a negative number.

Step 2: We need to check whether $y = -2$ is possible by solving $x^2 = -2$. In the real number system, no real number $x$ satisfies this equation since the square of any real number is non-negative.
Therefore, there is no real point where $y = -2$ on the graph of the function $y = x^2$.

Therefore, the solution to the problem is No.

To determine if the function $y = x^2$ passes through the point $(3, 36)$, we need to verify whether substituting $x = 3$ results in $y = 36$.

Substitute $x = 3$ into the function:

• The function is $y = x^2$.
• Substitute $x = 3$: $y = 3^2$
• Calculate the value: $y = 9$

The calculated value of $y$ when $x = 3$ is $9$. We compare this value to the given $y = 36$.

Since $9 \neq 36$, the function $y = x^2$ does not pass through the point $(3, 36)$.

Therefore, the correct answer is: No.

To solve this problem, we will calculate the value of $y$ for the given function $y = x^2$ when $x = 11$.

Here are the steps to find the solution:

• Step 1: Substitute $x = 11$ into the function equation.
• Step 2: Calculate $y = 11^2$.

Let's work through these steps:

Step 1: By substituting $x = 11$, the function becomes $y = 11^2$.

Step 2: Calculate the value:

$11^2 = 11 \times 11 = 121$

Therefore, for $x = 11$, the value of $y$ is $121$, meaning the point $(11, 121)$ does exist on the function $y = x^2$.

The correct answer from the given choices is $y = 121$, which corresponds to choice id "3".

The task is to find the value of $y$ when $x = 7$ for the function given by $y = x^2$.

To solve this problem, we will perform the following steps:

• Step 1: Substitute $x = 7$ into the function, resulting in the expression $y = 7^2$.
• Step 2: Calculate the square of 7, which is $7 \times 7 = 49$.
• Step 3: Hence, the value of $y$ is 49 when $x = 7$.

The calculation shows that the point exists on the parabola described by $y = x^2$ and is represented by the ordered pair $(7, 49)$.

Therefore, the solution to the problem, supported by calculations, is $y = 49$.