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).