Sure! Let’s solve these problems one by one.
2.1 To determine the x-intercepts of f, we need to find the values of x when f(x) = 0. So we need to solve the equation -2x² - 5x + 3 = 0. Using the quadratic formula, we get that x = (-(-5) ± √((-5)² - 4(-2)(3))) / (2(-2)). This simplifies to x = (5 ± √(49)) / (-4). So the x-intercepts are (5 - √(49)) / (-4) and (5 + √(49)) / (-4).
2.2 The axis of symmetry of a parabola in the form f(x) = ax² + bx + c is given by the formula x = -b / (2a). In this case, a = -2 and b = -5, so the axis of symmetry is x = -(-5) / (2(-2)) = 5/4.
2.3 The range of a quadratic function in the form f(x) = ax² + bx + c is determined by its vertex. The x-coordinate of the vertex is given by the formula x = -b / (2a), which we already calculated to be 5/4. The y-coordinate of the vertex is given by f(5/4) = -2(5/4)² - 5(5/4) + 3 = -25/8 - 25/4 + 3 = -25/8 - 50/8 + 24/8 = -51/8.
Since a < 0, the parabola opens downwards and the vertex is the highest point on the graph. Therefore, the range of f is (-∞, -51/8].
2.4 Reflecting the graph of f in the x-axis means that we change the sign of the y-coordinates. This is equivalent to replacing f(x) with -f(x). So the equation of the reflected graph is -f(x) = 2x² + 5x - 3.
Translating this graph 4 units to the right means that we replace x with (x - 4). So the equation of the translated graph is g(x) = 2(x - 4)² + 5(x - 4) - 3. Expanding this expression, we get g(x) = 2x² - 16x + 32 + 5x - 20 - 3 = 2x² - 11x + 9.
So the equation of g in the form g(x) = a(x + p)² + q is g(x) = 2(x - (-11/4))² + (9 - (121/8)).