Question - Quadratic Function Analysis
Solution:
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.
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