To analyze the function $f(x) = 2x^2 - 3x - 5$, we can follow these steps:

1. Identify the Vertex (Line of Symmetry)

The vertex of a quadratic function $ax^2 + bx + c$ occurs at: $x = -\frac{b}{2a}$ For our function:

• $a = 2$
• $b = -3$

Calculating the vertex: $x = -\frac{-3}{2 \cdot 2} = \frac{3}{4}$

2. Calculate the Function Value at the Vertex

To find the minimum value: $f\left(\frac{3}{4}\right) = 2\left(\frac{3}{4}\right)^2 - 3\left(\frac{3}{4}\right) - 5$ Calculating step-by-step: $f\left(\frac{3}{4}\right) = 2 \cdot \frac{9}{16} - \frac{9}{4} - 5$ $= \frac{18}{16} - \frac{36}{16} - \frac{80}{16} = \frac{18 - 36 - 80}{16} = \frac{-98}{16} = -\frac{49}{8}$

3. Roots of the Equation

To find the roots, we solve $2x^2 - 3x - 5 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Calculating the discriminant: $b^2 - 4ac = (-3)^2 - 4 \cdot 2 \cdot (-5) = 9 + 40 = 49$

Calculating the roots: $x = \frac{3 \pm \sqrt{49}}{2 \cdot 2} = \frac{3 \pm 7}{4}$ This gives: $x_1 = \frac{10}{4} = 2.5, \quad x_2 = \frac{-4}{4} = -1$

4. Summary of Results

• Line of Symmetry: $x = \frac{3}{4}$
• Minimum Value: $f\left(\frac{3}{4}\right) = -\frac{49}{8}$
• Roots: $x = 2.5$ and $x = -1$

5. Sketch of the Curve

To sketch the curve, plot the vertex, roots, and a few additional points, ensuring the parabola opens upwards (since $a = 2 > 0$).

Tips for Sketching:

• Plot the vertex at $\left(\frac{3}{4}, -\frac{49}{8}\right)$.
• Mark the roots at $x = -1$ and $x = 2.5$.
• Draw a symmetric curve around the line $x = \frac{3}{4}$.

Would you like any additional details or further assistance? Here are some related questions you might consider:

1. How does changing the coefficients $a$, $b$, and $c$ affect the graph of the quadratic?
2. What is the significance of the vertex in quadratic functions?
3. Can you explain how to complete the square for this quadratic?
4. How would the graph look if $a$ were negative?
5. What are the applications of quadratic functions in real life?

Tip: Always check the discriminant when finding roots to determine the number of real solutions!