This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
a) To find the equation of the straight line, we first calculate the slope (m) using the two given points (-2, -1) and (1, 4). Step 1: Calculate the slope (m). The formula for the slope is m = (y_2 - y_1)/(x_2 - x_1). Let (x_1, y_1) = (-2, -1) and (x_2, y_2) = (1, 4). m = (4 - (-1))/(1 - (-2)) = (4 + 1)/(1 + 2) = (5)/(3) Step 2: Use the point-slope form of a linear equation, y - y_1 = m(x - x_1), or the slope-intercept form, y = mx + c. We will use the slope-intercept form. Substitute the slope m = (5)/(3) and one of the points, for example (1, 4), into y = mx + c. 4 = (5)/(3)(1) + c 4 = (5)/(3) + c c = 4 - (5)/(3) c = (12)/(3) - (5)/(3) c = (7)/(3) Step 3: Write the equation of the line. Substitute m = (5)/(3) and c = (7)/(3) into y = mx + c. y = (5)/(3)x + (7)/(3) The equation of the line is y = (5)/(3)x + (7)/(3). b) To find the equation of the parabola, we use the vertex form of a quadratic function, y = a(x - h)^2 + k, where (h, k) is the vertex. Step 1: Identify the vertex and substitute it into the vertex form. From the graph, the vertex is (h, k) = (4, -1). y = a(x - 4)^2 - 1 Step 2: Use another point on the parabola to find the value of a. From the graph, the parabola appears to pass through the point (5, 0) on the x-axis. Substitute (x, y) = (5, 0) into the equation. 0 = a(5 - 4)^2 - 1 0 = a(1)^2 - 1 0 = a - 1 a = 1 Step 3: Write the equation of the parabola. Substitute a = 1 into the vertex form. y = 1(x - 4)^2 - 1 y = (x - 4)^2 - 1 The equation of the parabola is y = (x - 4)^2 - 1. Send me the next one 📸