b)	Draw the graph for the function:

y  = 4x2 – 4x – 3

by finding the following values -2 ≤ x ≤ 3:

i)	Find the y intercept						[2 Marks]

ii)	Find the x intercepts						[2 Marks]

iii)	Find the turning Point of the graph.				[2 Marks]

To graph the function \( y = 4x^2 - 4x - 3 \), we need to find the following values:

### i) Find the **y-intercept**:
The **y-intercept** occurs when \( x = 0 \). To find it, substitute \( x = 0 \) into the equation:

\[
y = 4(0)^2 - 4(0) - 3 = -3
\]

Thus, the **y-intercept** is \( (0, -3) \).

### ii) Find the **x-intercepts**:
The **x-intercepts** occur where \( y = 0 \). To find the x-intercepts, we solve the quadratic equation:

\[
4x^2 - 4x - 3 = 0
\]

We can solve this using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 4 \), \( b = -4 \), and \( c = -3 \).

Substitute these values into the formula:

\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(4)(-3)}}{2(4)}
\]
\[
x = \frac{4 \pm \sqrt{16 + 48}}{8}
\]
\[
x = \frac{4 \pm \sqrt{64}}{8}
\]
\[
x = \frac{4 \pm 8}{8}
\]

So the two solutions are:
\[
x = \frac{4 + 8}{8} = \frac{12}{8} = 1.5
\]
\[
x = \frac{4 - 8}{8} = \frac{-4}{8} = -0.5
\]

Thus, the **x-intercepts** are \( (-0.5, 0) \) and \( (1.5, 0) \).

### iii) Find the **turning point**:
The turning point (vertex) of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:

\[
x = \frac{-b}{2a}
\]
Substitute \( a = 4 \) and \( b = -4 \) into the formula:

\[
x = \frac{-(-4)}{2(4)} = \frac{4}{8} = 0.5
\]

To find the y-coordinate, substitute \( x = 0.5 \) back into the original equation:

\[
y = 4(0.5)^2 - 4(0.5) - 3
\]
\[
y = 4(0.25) - 2 - 3 = 1 - 2 - 3 = -4
\]

Thus, the **turning point** is \( (0.5, -4) \).

### Graphing the function:

- **y-intercept**: \( (0, -3) \)
- **x-intercepts**: \( (-0.5, 0) \), \( (1.5, 0) \)
- **Turning point**: \( (0.5, -4) \)

Would you like me to graph this for you? Let me know if you'd like any details or have any questions!

### 5 Related Questions:
1. How do we know the parabola opens upwards or downwards?
2. Can you explain how the quadratic formula helps in finding the x-intercepts?
3. Why is the turning point important in graphing quadratic functions?
4. How does changing the coefficients in a quadratic function affect its graph?
5. What other methods could be used to find the turning point of a quadratic function?

### Tip:
For any quadratic function \( y = ax^2 + bx + c \), if \( a > 0 \), the parabola opens upwards (minimum point); if \( a < 0 \), it opens downwards (maximum point).

Draw the graph for the function y = 4x^2 - 4x - 3 by finding the following values -2 ≤ x ≤ 3: i) Find the y-intercept [2 Marks] ii) Find the x intercepts [2 Marks] iii) Find the turning Point of the graph. [2 Marks]

Learn how to graph the quadratic function y = 4x^2 - 4x - 3 by finding the y-intercept, x-intercepts, and the turning point. This step-by-step solution covers key algebraic concepts such as the quadratic formula and vertex calculation, suitable for students in Grades 9-11.

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