Parabola of the form y=x²: Write a function that expresses..

To find the perimeter of the square with side length $a$, we use the standard formula for the perimeter of a square:

$P = 4 \times (\text{side length}) = 4a$

This straightforward multiplication tells us that the perimeter is four times the side length because a square has four equal sides.

Therefore, the function that expresses the perimeter of the square is:

$y = 4a$

Among the given choices, the correct expression corresponding to this calculation is: $y = 4a$.

Thus, the solution to this problem is $y = 4a$.

To determine the area of a square given its side length, we utilize the fundamental formula for the area of a square:

• Step 1: Recall that the area $A$ of a square with side length $a$ is given by the formula $A = a^2$.
• Step 2: Translate this into a function form, identifying the area as $y$.
• Step 3: Write the function expressing the area: $y = a^2$.

To conclude, the area of the square can be expressed as the function $y = a^2$.

Out of the provided choices, the correct expression for the area of the square as a function of the side length $a$ is Choice 4: $y = a^2$.

To solve the problem, we will calculate the area of the triangle within a square of side length $a$.

Since the triangle is formed by a diagonal of the square, its base and height are both equal to the side length $a$ of the square. Thus, base = $a$ and height = $a$.

Using the formula for the area of a triangle, we have:

$\text{Area of the triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times a = \frac{1}{2} \times a^2$

This simplifies to $\frac{a^2}{2}$.

Therefore, the function that expresses the area of the shaded triangle is $\frac{a^2}{2}$, matching the choice with the answer: $y=\frac{a^2}{2}$.

The solution to the problem is $y=\frac{a^2}{2}$.

To find the length of the diagonal of a square with side length $a$, we will use the Pythagorean theorem.

A diagonal divides a square into two congruent right-angled triangles. The legs of these triangles are the sides of the square, both with length $a$. The diagonal then serves as the hypotenuse.

According to the Pythagorean theorem, the hypotenuse $c$ of a right triangle with legs $a$ is found via:

• $c = \sqrt{a^2 + a^2}$

Simplifying inside the square root gives us:

• $c = \sqrt{2a^2}$

We can further simplify this expression:

• $c = \sqrt{2} \cdot \sqrt{a^2} = \sqrt{2}a$

Thus, the length of the diagonal is expressed by the function $y = \sqrt{2}a$.

In this problem, this solution corresponds to choice 1, which is $y = \sqrt{2}a$.

The chosen answer is correct as verified by the application of the Pythagorean theorem to a square.

Therefore, the correct function for the diagonal is $y = \sqrt{2}a$.

The goal of this problem is to find the function that represents the circumference of the circle when the radius is $a$.

The formula for the circumference of a circle is:

• $C = 2\pi r$

We are given that the radius $r$ is $a$.

Using the formula $C = 2\pi r$, substitute $a$ for $r$:

$C = 2\pi a$

Thus, the function that expresses the circumference of the circle is:

$y = 2\pi a$

Among the given choices, $y = 2\pi a$ corresponds to choice number 4.

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

• Identify the formula needed to compute the area of a circle.
• Substitute the given radius into the formula.
• Simplify the expression to match with the provided choices.

Now, let's work through the solution:

Step 1: The problem gives us the radius of the circle as $a$.

Step 2: We'll apply the formula for the area of a circle:
$A = \pi r^2$

Step 3: Substitute $r = a$:
$A = \pi a^2$

Therefore, the function expressing the area of the circle is $\boxed{y = \pi a^2}$.