The given problem in Japanese reads as follows:

「関数 $y = 4x^2$ について、$x$ の値が $-3$ から $a$ まで増加するときの変化の割合は 12 になります。$a$ の値を求めなさい。」

Translation: For the function $y = 4x^2$, when the value of $x$ increases from $-3$ to $a$, the rate of change is 12. Find the value of $a$.

Solution:

The rate of change of the function $y = 4x^2$ as $x$ changes from $x_1 = -3$ to $x_2 = a$ can be computed using the average rate of change formula:

$\text{Rate of change} = \frac{y(x_2) - y(x_1)}{x_2 - x_1}$

Substitute the given function $y = 4x^2$ and the points $x_1 = -3$ and $x_2 = a$:

$\text{Rate of change} = \frac{4a^2 - 4(-3)^2}{a - (-3)}$

Since $4(-3)^2 = 4 \times 9 = 36$, the equation becomes:

$\text{Rate of change} = \frac{4a^2 - 36}{a + 3}$

We are told that the rate of change is 12, so:

$\frac{4a^2 - 36}{a + 3} = 12$

Now, solve this equation for $a$.

1. Multiply both sides by $a + 3$ to eliminate the denominator:

$4a^2 - 36 = 12(a + 3)$

2. Expand the right-hand side:

$4a^2 - 36 = 12a + 36$

3. Move all terms to one side:

$4a^2 - 12a - 72 = 0$

4. Divide through by 4 to simplify:

$a^2 - 3a - 18 = 0$

5. Solve this quadratic equation using the quadratic formula:

$a = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-18)}}{2(1)}$

$a = \frac{3 \pm \sqrt{9 + 72}}{2}$

$a = \frac{3 \pm \sqrt{81}}{2}$

$a = \frac{3 \pm 9}{2}$

Thus, we get two solutions:

$a = \frac{3 + 9}{2} = 6 \quad \text{or} \quad a = \frac{3 - 9}{2} = -3$

Since the problem specifies that $x$ increases from $-3$, we discard $a = -3$, and the solution is:

$a = 6$

Final Answer:

The value of $a$ is $6$.

Would you like further details or have any questions? Here are some related questions:

1. What is the geometric interpretation of the average rate of change?
2. How does the quadratic formula work in solving $ax^2 + bx + c = 0$?
3. Can you explain how to check the solution to the quadratic equation step by step?
4. Why do we discard the negative solution $a = -3$ in this problem?
5. What is the significance of the rate of change in a quadratic function?

Tip: Always remember to carefully substitute values and simplify equations step by step when working with algebraic expressions.