The function you've provided is:

\[
y = 4x^2 - 5x
\]

This is a quadratic equation in the standard form \(y = ax^2 + bx + c\), where:
- \(a = 4\)
- \(b = -5\)
- \(c = 0\)

### Key Features of the Quadratic Function

1. **Vertex**: The vertex of a parabola \(y = ax^2 + bx + c\) can be found using the formula:

\[
x = -\frac{b}{2a}
\]

Substitute the values of \(a = 4\) and \(b = -5\):

\[
x = -\frac{-5}{2 \cdot 4} = \frac{5}{8}
\]

Now substitute this \(x\)-value back into the original equation to find the \(y\)-coordinate of the vertex:

\[
y = 4 \left( \frac{5}{8} \right)^2 - 5 \left( \frac{5}{8} \right)
\]

Simplify:

\[
y = 4 \cdot \frac{25}{64} - \frac{25}{8} = \frac{100}{64} - \frac{200}{64} = -\frac{100}{64} = -\frac{25}{16}
\]

So, the vertex of the parabola is at:

\[
\left( \frac{5}{8}, -\frac{25}{16} \right)
\]

2. **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex. Its equation is:

\[
x = \frac{5}{8}
\]

3. **Y-Intercept**: The y-intercept is the point where the graph crosses the y-axis, which occurs when \(x = 0\):

\[
y = 4(0)^2 - 5(0) = 0
\]

Thus, the y-intercept is at (0, 0).

4. **X-Intercepts**: To find the x-intercepts, set \(y = 0\) and solve for \(x\):

\[
4x^2 - 5x = 0
\]

Factor out \(x\):

\[
x(4x - 5) = 0
\]

So, \(x = 0\) or \(4x - 5 = 0\), which gives:

\[
x = \frac{5}{4}
\]

Therefore, the x-intercepts are at \((0, 0)\) and \(\left( \frac{5}{4}, 0 \right)\).

5. **Direction of the Parabola**: Since \(a = 4 > 0\), the parabola opens upwards.

### Summary

- **Vertex**: \(\left( \frac{5}{8}, -\frac{25}{16} \right)\)
- **Axis of Symmetry**: \(x = \frac{5}{8}\)
- **Y-Intercept**: \((0, 0)\)
- **X-Intercepts**: \((0, 0)\) and \(\left( \frac{5}{4}, 0 \right)\)
- **Direction**: Opens upwards

Would you like more details, or do you have any questions?

### Related Questions

1. How do you find the maximum or minimum value of a quadratic function?
2. What is the significance of the discriminant in a quadratic equation?
3. How do you solve a quadratic equation using the quadratic formula?
4. How can you graph a quadratic function using its vertex and intercepts?
5. What transformations can be applied to the graph of a quadratic function?

#### Tip
Always remember that the vertex form of a quadratic function provides an easy way to identify the vertex and to graph the parabola.

Explore how to solve and graph the quadratic equation y = 4x^2 - 5x with a step-by-step solution. Learn about key concepts such as vertex, axis of symmetry, and intercepts. Ideal for students in Grades 8-10.

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