Sketch the curve of fx =2xsquare -5x-5 showing the line of symmetry and minimum or maximum value and roots of the equation 

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

### 1. Determine the Line of Symmetry
For a quadratic function in the form \( ax^2 + bx + c \), the line of symmetry can be found using the formula:
\[
x = -\frac{b}{2a}
\]
In this case:
- \( a = 2 \)
- \( b = -5 \)

Calculating the line of symmetry:
\[
x = -\frac{-5}{2 \cdot 2} = \frac{5}{4} = 1.25
\]

### 2. Calculate the Vertex (Minimum or Maximum Value)
The vertex of a quadratic function gives the minimum or maximum value, depending on the direction of the parabola. Since \( a > 0 \), the parabola opens upwards, indicating a minimum value. We substitute \( x = \frac{5}{4} \) back into the function:
\[
f\left(\frac{5}{4}\right) = 2\left(\frac{5}{4}\right)^2 - 5\left(\frac{5}{4}\right) - 5
\]
Calculating:
\[
= 2 \cdot \frac{25}{16} - \frac{25}{4} - 5 = \frac{50}{16} - \frac{100}{16} - \frac{80}{16} = \frac{50 - 100 - 80}{16} = \frac{-130}{16} = -\frac{65}{8} \approx -8.125
\]

### 3. Find the Roots of the Equation
To find the roots, we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Substituting \( a = 2, b = -5, c = -5 \):
\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2} = \frac{5 \pm \sqrt{25 + 40}}{4} = \frac{5 \pm \sqrt{65}}{4}
\]
Thus, the roots are:
\[
x_1 = \frac{5 + \sqrt{65}}{4}, \quad x_2 = \frac{5 - \sqrt{65}}{4}
\]

### Summary of Key Points
- **Line of Symmetry**: \( x = 1.25 \)
- **Minimum Value**: \( f\left(\frac{5}{4}\right) \approx -8.125 \)
- **Roots**: \( x_1 = \frac{5 + \sqrt{65}}{4}, \quad x_2 = \frac{5 - \sqrt{65}}{4} \)

### Sketch of the Curve
To visualize the curve, we would plot the parabola opening upwards, mark the line of symmetry at \( x = 1.25 \), the vertex at approximately \( (1.25, -8.125) \), and the roots along the x-axis.

Would you like to see a graphical representation of this curve? Or do you have any specific questions about this function? 

### Additional Questions to Explore:
1. How do the coefficients affect the shape of the parabola?
2. What happens to the vertex if the function is shifted vertically?
3. How can you find the y-intercept of the function?
4. What would the graph look like if \( a < 0 \)?
5. How do you derive the roots using factoring, if possible? 

**Tip**: Always sketch the graph to visualize key points like symmetry, maximum/minimum, and roots effectively!

Sketch the curve of fx =2xsquare -5x-5 showing the line of symmetry and minimum or maximum value and roots of the equation.

Explore how to sketch the curve of the quadratic function 2x² - 5x - 5, including the line of symmetry at x = 1.25, the minimum value at approximately -8.125, and the roots using the quadratic formula. Ideal for Grades 7-9 students.

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