Parabola of the Form y=(x-p)²: Finding a stationary point

To solve this problem, we will find the intersection of the function with the Y-axis by following these steps:

• Step 1: Recognize that the intersection with the Y-axis occurs where $x = 0$.
• Step 2: Substitute $x = 0$ into the function $y = (x+4)^2$.
• Step 3: Perform the calculation to find the y-coordinate.

Now, let's solve the problem:

Step 1: Identify the Y-axis intersection by setting $x = 0$.
Step 2: Substitute $x = 0$ into the function:

$y = (0+4)^2 = 4^2 = 16$

Step 3: The intersection point on the Y-axis is $(0, 16)$.

Therefore, the solution to the problem is $(0, 16)$.

To solve this problem, we'll find the intersection of the function $y = (x-2)^2$ with the x-axis. The x-axis is characterized by $y = 0$. Hence, we set $(x-2)^2 = 0$ and solve for $x$.

Let's follow these steps:

• Step 1: Set the function equal to zero:

$(x-2)^2 = 0$

• Step 2: Solve the equation for $x$:

Taking the square root of both sides gives $x - 2 = 0$.

Adding 2 to both sides results in $x = 2$.

• Step 3: Find the intersection point coordinates:

The x-coordinate is $x = 2$, and since it intersects the x-axis, the y-coordinate is $y = 0$.

Therefore, the intersection point of the function with the x-axis is $(2, 0)$.

The correct choice from the provided options is $(2, 0)$.

To determine the intersection of the function $y = (x-2)^2$ with the y-axis, we set $x = 0$, as the y-axis is defined by all points where $x = 0$.

Substituting $x = 0$ into the equation:

$y = (0 - 2)^2$

Simplifying this expression:

$y = (-2)^2 = 4$

Thus, the intersection point of the function with the y-axis is $(0, 4)$.

Therefore, the solution to the problem is $(0, 4)$.

To find the intersection of the function $y = (x-6)^2$ with the y-axis, we follow these steps:

• Step 1: Identify the known function and approach the problem by setting $x = 0$ since we are looking for the intersection with the y-axis.

• Step 2: Substitute $x = 0$ into the equation $y = (x-6)^2$.

• Step 3: Perform the calculation to find $y$.

Now, execute these steps:
Step 1: We are given the function $y = (x-6)^2$.
Step 2: Substitute $x = 0$ into the equation:
$y = (0-6)^2$
Step 3: Simplify the expression:
$y = (-6)^2 = 36$

The point of intersection with the y-axis is therefore $(0, 36)$.

Thus, the solution to the problem is $(0, 36)$.

To find the intersection of the parabola $y = (x+3)^2$ with the y-axis, we set $x = 0$ since any point on the y-axis has its x-coordinate as zero.

Step-by-step solution:

• Step 1: Substitute $x = 0$ into the equation: $y = (0+3)^2$.
• Step 2: Simplify the expression:
• $y = (3)^2$
• $y = 9$

Therefore, the intersection point of the parabola with the y-axis is $(0, 9)$.

Accordingly, among the given choices, the correct choice for the intersection is $(0, 9)$.

To find the intersection of the function $y = (x-5)^2$ with the y-axis, we follow these steps:

• Step 1: Substitute $x = 0$ into the function to find the y-intercept.

Let's calculate:
- Substitute $x = 0$ into the function: $y = (0-5)^2$.
- Simplifying further, $y = (-5)^2 = 25$.

Thus, the intersection of the function $y = (x-5)^2$ with the y-axis occurs at the point $(0, 25)$.

Therefore, the correct answer is $(0, 25)$.

To solve this problem, we will follow these steps:

• Step 1: Identify the function as $y = (x+3)^2$.
• Step 2: Set $y = 0$ to find the intersection with the x-axis.
• Step 3: Solve $(x+3)^2 = 0$ to find the x-coordinate.

Let's proceed through each step:
Step 1: We are given the function $y = (x+3)^2$.
Step 2: To find where this function intersects the x-axis, set $y = 0$:

$(x+3)^2 = 0$

Step 3: Solve the equation:
The equation $(x+3)^2 = 0$ suggests that $x+3 = 0$,
which simplifies to $x = -3$.

Therefore, the intersection point is where $x = -3$ and $y = 0$, giving us the intersection at $(-3, 0)$.

Thus, the solution to the problem is $(-3, 0)$, corresponding to choice given as $(-3, 0)$.

To find the intersection of the parabola $y = (x-5)^2$ with the x-axis, we must set the value of $y$ to zero since any point on the x-axis has a y-coordinate of zero.

Solving the equation:

• Set $(x-5)^2 = 0$.
• To solve for $x$, remove the square by taking the square root of both sides. This yields $x-5 = 0$.
• Solve for $x$ by adding 5 to both sides: $x = 5$.

Thus, the intersection point of the parabola with the x-axis is at $(5, 0)$.

Therefore, the correct answer is $\boxed{(5,0)}$, which corresponds to choice 3.

To find the intersection of the parabola $y = (x - \frac{1}{2})^2$ with the x-axis, we need to set $y = 0$ because at the x-axis, the y-coordinate is always zero.

Let's go through the steps:

• Step 1: Set the equation equal to zero: $(x - \frac{1}{2})^2 = 0$.
• Step 2: Solve for $x$. A square equals zero only when the base itself is zero, thus:
  $x - \frac{1}{2} = 0$
• Step 3: Solve this equation for $x$:
  $x = \frac{1}{2}$

The intersection point on the x-axis has coordinates $(x, y)$, where we have found $x = \frac{1}{2}$ and we know $y = 0$.

Therefore, the intersection of the function with the x-axis is at the point $(\frac{1}{2}, 0)$.

Thus, the correct answer is choice 3: $(\frac{1}{2}, 0)$.

To find the intersection of the function $y = (x + 1\frac{1}{4})^2$ with the x-axis, we set $y = 0$ since intersections on the x-axis have a $y$-coordinate of zero.

Therefore, our equation becomes:

$(x + 1\frac{1}{4})^2 = 0$.

To solve this equation, take the square root of both sides:

$x + 1\frac{1}{4} = 0$.

Next, solve for $x$ by subtracting $1\frac{1}{4}$ from both sides:

$x = -1\frac{1}{4}$.

Thus, the intersection point of the function with the x-axis is $(-1\frac{1}{4}, 0)$.

After checking the provided answer choices, the correct choice is:

: $(-1\frac{1}{4},0)$

Therefore, the solution to the problem is $(-1\frac{1}{4}, 0)$.