91Ó°ÊÓ

reflection in the \(y\)-axis followed by a translation 4 units right $$ \begin{aligned} h(x) &=f(-x) \\ &=2(-x)^2+6(-x) \\ &=2(x)^2-6 x \\ g(x) &=h(x-4) \\ &=2(x-4)^2+6(x-4) \\ &=2 x^2-10 x+8 \end{aligned} $$

Identify the vertex, focus, directrix, and axis of symmetry of the parabola. Describe the transformations of the graph of the standard equation with vertex \((\mathbf{0}, \mathbf{0})\). $$y=(x+3)^2-5$$

MODELING WITH MATHEMATICS The function f(t) = ?16t2 + 10 models the height (in feet) of an object t seconds after it is dropped from a height of 10 feet on Earth. The same object dropped from the same height on the moon is modeled by g(t) = ? 8 —3 t2 + 10. Describe the transformation of the graph of f to obtain g. From what height must the object be dropped on the moon so it hits the ground at the same time as on Earth?

MODELING WITH MATHEMATICS Flying fi sh use their pectoral fi ns like airplane wings to glide through the air. a. Write an equation of the form y = a(x ? h)2 + k with vertex (33, 5) that models the fl ight path, assuming the fi sh leaves the water at (0, 0). b. What are the domain and range of the function? What do they represent in this situation? c. Does the value of a change when the fl ight path has vertex (30, 4)? Justify your answer.

THOUGHT PROVOKING A jump on a pogo stick with a conventional spring can be modeled by f(x) = ?0.5(x ? 6)2 + 18, where x is the horizontal distance (in inches) and f(x) is the vertical distance (in inches). Write at least one transformation of the function and provide a possible reason for your transformation.

COMPARING METHODS Let the graph of g be a translation 3 units up and 1 unit right followed by a vertical stretch by a factor of 2 of the graph of f(x) = x2. a. Identify the values of a, h, and k and use vertex form to write the transformed function. b. Use function notation to write the transformed function. Compare this function with your function in part (a). c. Suppose the vertical stretch was performed fi rst, followed by the translations. Repeat parts (a) and (b). d. Which method do you prefer when writing a transformed function? Explain.

As \(|p|\) increases, how does the width of the graph of the equation \(y=\frac{1}{4 p} x^2\) change? Explain your reasoning.

PROBLEM SOLVING The path of a diver is modeled by the function f(x) = ?9x2 + 9x + 1, where f(x) is the height of the diver (in meters) above the water and x is the horizontal distance (in meters) from the end of the diving board. a. What is the height of the diving board? b. What is the maximum height of the diver? c. Describe where the diver is ascending and where the diver is descending.

Use the Distance Formula to derive the equation of a parabola that opens to the right with vertex \((0,0)\), focus \((p, 0)\), and directrix \(x=-p\).

REASONING The points (2, 3) and (?4, 2) lie on the graph of a quadratic function. Determine whether you can use these points to fi nd the axis of symmetry. If not, explain. If so, write the equation of the axis of symmetry