https://file.aiquickdraw.com/mathbot/file/1726366667847lo6v23ph.jpg

Let's work through the three sets of questions in the image:

### 1. **Soal Materi: Fungsi Kuadrat**
The quadratic function is \( f(x) = 2ax^2 - 4ax + 8a \). The problem asks under which conditions the function is always increasing.

For a quadratic function \( f(x) = Ax^2 + Bx + C \), the function is increasing for all \( x \) if the leading coefficient \( A \) is positive and the vertex of the parabola is such that the function is always increasing beyond it.

Given:
\[ f(x) = 2ax^2 - 4ax + 8a \]
Here, \( A = 2a \).

1. If \( a > 0 \), the parabola opens upwards, and it is increasing if the discriminant \( \Delta \) of the quadratic equation is negative (no real roots, meaning the function never crosses the x-axis). 
   \[
   \Delta = (-4a)^2 - 4(2a)(8a) = 16a^2 - 64a^2 = -48a^2
   \]
   For \( a > 0 \), the discriminant is negative, so the function is always increasing.

2. If \( a < 0 \), the parabola opens downwards, and for the same reason (negative discriminant), the function is always increasing when \( a < 0 \).

Thus, the correct answer is **C**: Jika \( a < 0 \), maka \( x > 1 \).

---

### 2. **Soal Materi: Persamaan Linear**
We are given two linear equations:
\[
4x - 4y = 3 - p
\]
\[
6x - 8y = 1
\]
We need to find the value of \( p \) such that \( x \) and \( y \) are positive integers, and \( p > 1 \). Also, \( p - 4x \) is to be determined.

1. Solve the system of linear equations to find \( x \) and \( y \).
2. Substitute these into the given conditions to find \( p \).

---

### 3. **Soal Materi: Limit**
The limit to evaluate is:
\[
\lim_{x \to y} \frac{2x^2 + 4xy - 6y^2}{x - y}
\]
Factor the numerator:
\[
2x^2 + 4xy - 6y^2 = 2(x^2 + 2xy - 3y^2)
\]
Factor further:
\[
2(x - y)(x + 3y)
\]
Now, cancel the \( x - y \) term:
\[
\lim_{x \to y} \frac{2(x - y)(x + 3y)}{x - y} = \lim_{x \to y} 2(x + 3y)
\]
When \( x = y \), this becomes:
\[
2(4y) = 8y
\]
Thus, the correct answer is **E: 8y**.

---

Would you like further details on any specific part of the solution? Here are 5 related questions you might explore:

1. How do you factor complex quadratic expressions?
2. How do discriminants help determine the nature of roots in quadratic functions?
3. What methods can you use to solve systems of linear equations?
4. How do you simplify limits involving factoring and cancellation?
5. How does changing the sign of a leading coefficient affect the shape of a parabola?

**Tip:** When working with limits that involve factoring, always check for possible common terms you can cancel to simplify the expression.

The user has submitted an image containing three math problems: one on quadratic functions, one on linear equations, and one on limits.

This page provides solutions to three high school-level math problems involving quadratic functions, linear equations, and limits. Learn how to solve quadratic inequalities, solve systems of linear equations, and evaluate limits step-by-step.

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