Exam 9: Conics, Systems of Nonlinear Equations and Inequalities, and Parametric Equations

(a) Identify the type of conic by the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY term) by rotating the x- and y-axes by an angle of θ to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).

Find an equation for the parabola with focus (3, 12) and directrix y = 4.

Find the standard form of the equation of an ellipse with foci (0, -2) and (0, 2) and vertices (0, -5) and (0, 5).

Recall that the flight of a projectile can be modeled with the parametric equations: , where t is in seconds, v_o is the initial velocity, $\theta$ is the angle with the horizontal, and x and y are in feet. A baseball is hit at an initial speed of 100 mph and angle of 32° at a height of 3 feet above the ground. If home plate is 465 feet from the back fence, which is 19 feet tall, will the baseball clear the back fence for a home run?

Graph the nonlinear inequality.

Identify the type of conic section shown in the graph.

For the polar equation, graph the conic.

Find the points of intersection of the two parabolas. Parabola I: vertex (1, -10); focus: (1, -9.142857) Parabola II: vertex (1, 11); focus: (1, 10.142857)

Find the polar equation that represents the conic described (assume the focus is at the origin). Hyperbola with eccentricity e = and directrix

Identify the conic (parabola, ellipse, or hyperbola) that is represented by the equation.

Find the standard form of the equation of a hyperbola with center (0, 0), transverse x-axis, and asymptotes And

Recall that the flight of a projectile can be modeled with the parametric equations: , where t is in seconds, v_o is the initial velocity, $\theta$ is the angle with the horizontal, and x and y are in feet. A missile is fired from a ship at an angle of 39°, an initial height of 20 feet above the water's surface, and at a speed of 3890 feet per second. Will it be able to hit a target that is 2.4 miles away?

Find the standard form of the equation of a hyperbola with center (0, 0), transverse y-axis, and asymptotes y = -7x and y = 7x.

Find the standard form of the equation of an ellipse with foci (-3, 0) and (3, 0) and vertices (-4, 0) and (4, 0).

Graph the hyperbola.

Solve the system of equations by applying the substitution method. Y = x² - 3 Y = 3x - 3

Match the graph to the system of inequalities.

Graph the ellipse.

Recall that the flight of a projectile can be modeled with the parametric equations: , where t is in seconds, ν_o is the initial velocity, θ is the angle with the horizontal, and x and y are in feet. A projectile is launched at a speed of 200 ft/sec at an angle of 30° with the horizontal. Plot the path of the projectile on a graph.

(a) Identify the type of conic by the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY term) by rotating the x- and y-axes by an angle of θ to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).