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Encontar domino 

Interacciones en x y y

Construcción gráfica 

Rango o imagen de la función

Let's analyze the provided exercise step by step. The task involves evaluating the function \( f(x) = 3x^2 - 5x \).

### 1. Finding the Domain
The domain of a polynomial function like \( f(x) = 3x^2 - 5x \) is all real numbers since there are no restrictions (like division by zero or square roots of negative numbers) in the function. Therefore:
\[ \text{Domain: } (-\infty, \infty) \]

### 2. Finding the Intercepts
#### x-intercepts:
To find the x-intercepts, set \( f(x) = 0 \):
\[ 3x^2 - 5x = 0 \]
\[ x(3x - 5) = 0 \]
Thus, the solutions are:
\[ x = 0 \]
\[ 3x - 5 = 0 \Rightarrow x = \frac{5}{3} \]

The x-intercepts are \( (0, 0) \) and \( \left(\frac{5}{3}, 0\right) \).

#### y-intercept:
To find the y-intercept, set \( x = 0 \):
\[ f(0) = 3(0)^2 - 5(0) = 0 \]

The y-intercept is \( (0, 0) \).

### 3. Constructing the Graph
To graph \( f(x) = 3x^2 - 5x \):
1. Plot the intercepts: \( (0, 0) \) and \( \left(\frac{5}{3}, 0\right) \).
2. Identify the vertex of the parabola. For \( f(x) = ax^2 + bx + c \), the vertex occurs at \( x = -\frac{b}{2a} \):
   \[ x = -\frac{-5}{2 \cdot 3} = \frac{5}{6} \]
   \[ f\left(\frac{5}{6}\right) = 3\left(\frac{5}{6}\right)^2 - 5\left(\frac{5}{6}\right) = \frac{75}{36} - \frac{150}{36} = -\frac{75}{36} = -\frac{25}{12} \]
   So the vertex is \( \left(\frac{5}{6}, -\frac{25}{12}\right) \).

3. Sketch the parabola opening upwards with vertex at \( \left(\frac{5}{6}, -\frac{25}{12}\right) \) and passing through the intercepts.

### 4. Finding the Range
Since the parabola opens upwards (the coefficient of \( x^2 \) is positive), the range of \( f(x) \) starts from the minimum value at the vertex and goes to \( \infty \):
\[ \text{Range: } \left[-\frac{25}{12}, \infty\right) \]

### Summary
- **Domain:** \( (-\infty, \infty) \)
- **x-intercepts:** \( (0, 0) \) and \( \left(\frac{5}{3}, 0\right) \)
- **y-intercept:** \( (0, 0) \)
- **Vertex:** \( \left(\frac{5}{6}, -\frac{25}{12}\right) \)
- **Range:** \( \left[-\frac{25}{12}, \infty\right) \)

Do you want more details on any of these steps, or have any questions? Here are some additional related questions:

1. How do we derive the general form of the quadratic function?
2. What is the significance of the vertex in a quadratic function?
3. How do we determine the axis of symmetry for the parabola?
4. Can a quadratic function have no real x-intercepts? Under what conditions?
5. How does the leading coefficient of the quadratic function affect the shape of the parabola?
6. What are the methods for solving quadratic equations?
7. How do we find the maximum or minimum value of a quadratic function?
8. How do transformations affect the graph of a quadratic function?

**Tip:** When graphing a function, always identify key features like intercepts and the vertex to help accurately sketch the curve.

Explore how to find domain, intercepts, construct the graph, and determine the range of the quadratic function f(x) = 3x^2 - 5x. This detailed solution covers key algebraic concepts and graphing techniques for Grades 10-12 students.

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