Parabola of the form y=x² - Examples, Exercises and Solutions

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

• Step 1: Substitute the given value of $x$ into the equation.
• Step 2: Perform the calculation to find $y$.

Now, let's work through each step:
Step 1: The given equation is $y = x^2$. We need to substitute $x = 2$ into this equation.

Step 2: Substitute to get $y = (2)^2$. Calculate $2 \times 2 = 4$.

Therefore, the value of $y$ when $x = 2$ is $y = 4$.

Hence, the solution to the problem is $y = 4$.

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

• Step 1: Set up the equation from the function definition.
• Step 2: Solve the equation by taking the square root of both sides.
• Step 3: Identify all possible values for $x$.
• Step 4: Compare with the given answer choices.

Now, let's work through each step:

Step 1: We start with the equation given by the function $f(x) = x^2$. We know $f(?) = 16$, so we can write:

$x^2 = 16$

Step 2: To solve for $x$, we take the square root of both sides of the equation:

$x = \pm \sqrt{16}$

Step 3: Solve for $\sqrt{16}$:

The square root of 16 is 4, so:

$x = 4$ or $x = -4$

This gives us the two solutions: $x = 4$ and $x = -4$.

Step 4: Compare these solutions to the answer choices. The correct choice is:

$f(4)$ and $f(-4)$

Therefore, the solution to the problem is $f(4)$ and $f(-4)$.

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

• Step 1: Set the equation of the function with the given point, $x^2 = 4$.
• Step 2: Solve for $x$ by taking the square root of both sides. This accounts for both the positive and negative solutions.
• Step 3: Evaluate the expression to find the solutions.

Now, let's work through each step:
Step 1: Set up the equation based on the given information:
We have $x^2 = 4$.

Step 2: Solve by taking the square root of both sides:
Taking the square root, we get $x = \pm\sqrt{4}$.

Step 3: Simplify to find the values of $x$:
The square root of 4 is 2, thus $x = 2$ and $x = -2$.

Therefore, the solutions for $x$ are $x = 2$ and $x = -2$.

The correct answer is choice Answers a + b, which corresponds to having solutions $x = 2$ and $x = -2$.

To solve this problem, let's find the steps required to determine $x$ when $y = 16$ in the function $y = x^2$:

• Step 1: Substitute the given $y$ into the equation to get $x^2 = 16$.
• Step 2: To solve $x^2 = 16$, take the square root of both sides, remembering to include both positive and negative roots. This yields $x = \pm\sqrt{16}$.
• Step 3: Simplify to find $x = \pm4$, which gives the solutions $x = 4$ and $x = -4$.

Thus, the value(s) of $x$ that satisfy $y = 16$ in the function $y = x^2$ are $x = 4$ and $x = -4$.

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

To solve the problem, we will proceed with the following steps:

• Identify the provided equation and condition.
• Apply the square root property to solve the equation.
• Verify the solution with the given choices.

Step-by-step solution:

Step 1: Substitute $y = 36$ into the equation $y = x^2$, which gives:

$x^2 = 36$

Step 2: Solve for $x$ by taking the square root of both sides. Using the square root property, we have:

$x = \pm \sqrt{36}$

Since the square root of 36 is 6, we find that:

$x = \pm 6$

Therefore, the solutions to the equation are $x = 6$ and $x = -6$.

Thus, the value of $x$ for $y = 36$ in the function $y = x^2$ is $x = \pm 6$.

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

• Step 1: Identify the value given for $x$.
• Step 2: Substitute the given $x$ value into the function.
• Step 3: Calculate the resulting value for $y$.

Now, let's work through each step:
Step 1: The problem states that $x = 6$.
Step 2: Using the function $y = x^2$, we substitute $x = 6$.
Step 3: Perform the calculation: $y = 6^2$.

Calculating $6^2$, we get $36$.
Therefore, for the function $y = x^2$, when $x = 6$, the value of $y$ is $y = 36$.

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 solve this problem, we'll follow these steps:

• Step 1: Set up the equation $x^2 = 9$.
• Step 2: Solve for $x$ by taking the square root of both sides.
• Step 3: Choose the correct answer from the given options.

Now, let's work through each step:
Step 1: We set $x^2 = 9$.
Step 2: Solving for $x$, we take the square root of both sides: $x = \pm \sqrt{9}$.
Step 3: This yields two solutions: $x = 3$ and $x = -3$.

Comparing these values with the given choices:

• Choice 1: $f(3)$ corresponds to $x = 3$.
• Choice 3: $f(-3)$ corresponds to $x = -3$.

Both choices $f(3)$ and $f(-3)$ are correct, leading us to select the combined choice: Answer A+C.

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

• Step 1: Set up the equation based on the function $f(x)=x^2$ for $f(?)=25$.
• Step 2: Solve for $x$ by applying the square root operation.

Now, let's work through each step:
Step 1: We start with the equation $x^2 = 25$ derived from $f(x) = 25$.
Step 2: To solve for $x$, we take the square root of both sides:

$x = \pm \sqrt{25}$

Calculating the square root gives us $x = \pm 5$. However, we are looking for a specific point that fits one of the answer choices:
Therefore, the solution based on the choices provided is $x = 5$.

Concluding, the missing value of the function point is $f(5)$, which coincides with choice 1.

Let's solve the problem by following these steps:

• Step 1: Identify the equation to solve.
• Step 2: Apply the square root to both sides of the equation.
• Step 3: Solve for both positive and negative values of $x$.

Step 1: We start with the equation derived from the function:
$x^2 = 25$

Step 2: To isolate $x$, we take the square root of both sides. Remember, the square root of a number can be both positive and negative:
$x = \pm \sqrt{25}$

Step 3: Simplify the square root:
$x = \pm 5$, which means $x = 5$ or $x = -5$

Therefore, the values of $x$ that satisfy $y = 25$ in the function $y = x^2$ are $x = 5$ and $x = -5$.

Looking at the choices given, the correct answer is:

$x=5,x=-5$

The problem requires us to find the value of $x$ for the function $y = x^2$ when $y = 8$.

Let's solve this step-by-step:

• Step 1: Set up the equation based on the given function:
  $\ y = x^2$ becomes $\ x^2 = 8$.
• Step 2: Solve for $x$ by taking the square root of both sides:

Since $\ x^2 = 8$, we take the square root of both sides to find $x$:

$x = \pm\sqrt{8}$

Step 3: Simplify the square root:

The square root of 8 can be simplified to $\ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$.

Thus, $\ x = \pm 2\sqrt{2}$.

Therefore, the solution is $x = \pm 2\sqrt{2}$, which corresponds to choice 4.