Nonlinear functions：練習問題

If the function \( f \) is defined by \( f(x) = x^2 - 4x + 7 \), what is the value of \( f(3) \)?

The function \( f \) is defined by \( f(x) = x^2 + bx + 7 \), where \( b \) is a constant. If \( f(3) = 1 \), what is the value of \( b \)?

The graph of a quadratic function \( h(x) = a(x-h)^2 + k \) in the \( xy \)-plane has its vertex at \( (3, 2) \) and passes through the point \( (1, -6) \). What is the value of \( h(5) \)?

If the function \( g \) is defined by \( g(x) = 3^x - 5 \), what is the value of \( g(2) \)?

The graph of the quadratic function \( f(x) = a(x - 2)^2 + 5 \) is shown in the \( xy \)-plane. If the graph passes through the point \( (4, 13) \), what is the value of \( a \)?

A biologist models the population of a certain bacteria culture using the function \( P(t) = 500(2)^{\frac{t}{3}} \), where \( P(t) \) is the number of bacteria after \( t \) hours. After how many hours will the population reach 4,000?

In the \( xy \)-plane, the graph of the function \( f(x) = x^2 - 4x - 12 \) intersects the \( x \)-axis at points \( A \) and \( B \). What is the distance between point \( A \) and point \( B \)?

A ball is thrown upward from a platform. Its height \( h \), in meters, above the ground \( t \) seconds after it is thrown is modeled by the function \( h(t) = -5t^2 + 20t + 10 \).
a) What is the initial height, in meters, of the platform from which the ball was thrown?
b) At what time \( t \), in seconds, does the ball reach its maximum height above the ground?
c) What is the maximum height, in meters, reached by the ball?
d) Determine the time \( t \) when the ball hits the ground, rounding your answer to the nearest hundredth if necessary.