Example Question - quadratic function analysis
Here are examples of questions we've helped users solve.
Interpreting a Quadratic Graph
The image displays a graph with a curve, but unfortunately, the question you're asking for help with is not visible in the image. I can, however, provide some general observations about the graph: - The curve appears to be a plot of a quadratic function, which is typically expressed in the form \( y = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants. - The graph indicates that the quadratic function has one real root (zero) where the curve crosses the x-axis. - The vertex of the parabola (the highest or lowest point on the curve) is located in the second quadrant of the Cartesian coordinate system, implying the "a" coefficient in the quadratic equation is negative, as the parabola opens downward. - The y-intercept (the point where the curve crosses the y-axis) is above the origin, suggesting that the "c" term in the equation is positive. To provide a more specific solution or analysis, I would need to know the actual question or the function that was used to generate this curve. If you wanted to find the equation of the curve, you would require either its vertex and one other point or the y-intercept and two other points on the curve. If the question is to analyze characteristics such as intercepts, vertex, axis of symmetry, maxima/minima, or intervals of increase/decrease, please provide those details.
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.
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