Given the following function, (a) find the vertex; (b) determine whether there is a maximum or a minimum value, and find the value; (c) find the range; and (d) find the intervals on which the function is increasing and the intervals on which the function is decreasing. \( f(x)=3 x^{2}+6 x-45 \) Select the correct choice below and fill in the answer box within your choice. (Type an integer or a fraction.) A. The parabla opens downward and has a maximum value of B. The parabola opens upward and has a minimum value of (c) What is the range of \( f(x) \) ? The range of \( f(x) \) is \( \square \). (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) (d) On what interval is the function increasing? The function is increasing on \( \square \). (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)

The table contains values for \( f(x) \). If \( h(x) \) is the reflection of \( f(x) \) across the \( x \)-axis, find the missing value in the ordered pair below, for \( h(x) \). \( x \quad-\frac{\pi}{2}-\frac{\pi}{4} \) \( f(x) \) 0

Use transformations to graph the function. Determine the domain, range, horizontal asymptote, and \( y \)-intercept of the function. \( f(x)=4^{x}+3 \) Use the graphing tool to graph the function. What is the domain of \( f(x)=4^{x}+3 \) ? (For any answer boxes shown with the grapher, type an exact answer.) (Type your answer in interval notation.)

Use the theorems proved in class about induction to evaluate the sums below. (Note that you are begin asked to apply the theorems, not formally proving a new theorem.) a. \( 7+14+21+28+\cdots+350^{\circ} \) b. \( 16+17+18+\cdots+411 \) c. \( 1-\frac{1}{10}+\frac{1}{100}-\frac{1}{1000} \cdots+\frac{1}{10^{18}} \)

Let \( f(x)=7\left(\frac{1}{2}\right)^{x} \). Complete parts (a) through (e) below (c) Find any asymptotes of the graph of \( f \). Select the co

Graph all vertical and horizontal asymptotes of the \[ f(x)=\frac{8 x-7}{-2 x-1} \]