Quadratic Function Analysis
The image shows a question with two parts, (i) and (ii), concerning the graph of the function \(y = -3x^2 + 4x + 3\).
(i) **Find the intercepts and the coordinates of the turning point on the graph of** \(y = -3x^2 + 4x + 3\).
**To find the x-intercepts**, we set \(y\) equal to zero and solve for \(x\):
\[0 = -3x^2 + 4x + 3\]
This is a quadratic equation, and we can solve it using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -3\), \(b = 4\), and \(c = 3\). Let's solve for \(x\):
\(x = \frac{-4 \pm \sqrt{4^2 - 4(-3)(3)}}{2(-3)}\)
\(x = \frac{-4 \pm \sqrt{16 + 36}}{-6}\)
\(x = \frac{-4 \pm \sqrt{52}}{-6}\)
\(x = \frac{-4 \pm 2\sqrt{13}}{-6}\)
Since we cannot simplify the square root any further, we have:
\(x_1 = \frac{-4 + 2\sqrt{13}}{-6}\) and \(x_2 = \frac{-4 - 2\sqrt{13}}{-6}\)
**To find the y-intercept**, we set \(x\) equal to zero:
\(y = -3(0)^2 + 4(0) + 3 = 3\)
So the y-intercept is at the point (0, 3).
**To find the turning point** (also known as the vertex), we can use the formula for the x-coordinate of the vertex, \(x = -\frac{b}{2a}\), where \(a = -3\) and \(b = 4\):
\(x = -\frac{4}{2(-3)}\)
\(x = \frac{4}{6}\)
\(x = \frac{2}{3}\)
Now substitute \(x = \frac{2}{3}\) into the equation to find the y-coordinate:
\(y = -3\left(\frac{2}{3}\right)^2 + 4\left(\frac{2}{3}\right) + 3\)
\(y = -3\left(\frac{4}{9}\right) + \frac{8}{3} + 3\)
\(y = -\frac{4}{3} + \frac{8}{3} + 3\)
\(y = \frac{4}{3} + 3\)
\(y = \frac{4}{3} + \frac{9}{3}\)
\(y = \frac{13}{3}\)
So the coordinates of the turning point are \( \left(\frac{2}{3}, \frac{13}{3}\right) \).
(ii) **Use the graph to explain why** \(y = -3x^2 + 4x + 3\) **has two distinct real roots.**
The question is asking to use the graph as an explanation, but since we cannot see the graph, I will provide a mathematical explanation based on what we have calculated.
The quadratic equation \(y = -3x^2 + 4x + 3\) has two distinct real roots because the square root part of the quadratic formula resulted in \(\sqrt{52}\), which is a real number (as all square roots of positive numbers are real). Having two distinct roots means that the discriminant \(b^2 - 4ac\) is positive (\(52\) in our case), and this leads to two different x-intercepts on the graph, which we calculated earlier. In the context of a graph, this means that the parabola crosses the x-axis at two points, corresponding to the two distinct real roots.
