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Find the equation of the parabola \( y = 2 x^2 + b x + c\) that passes through the points \( (-1,-5)\) and \( (2,10)\). What is the equation of the parabola with x intercepts at \( x = 2\) and \( x = -3\), and a y - intercept at \( y = 5\)?
Stan Ulam that graph of any quadratic function can be obtained from the core parabola, f~x! 5 x2, by applying basic transformations. We apply terminology from the core parabola to parabolas in general. The point (0, 0) is called thevertex of the core parabola, and they-axis is the axis of symmetry. The axis of symmetry is a help
equation has linear terms -2hx and -2ky-they disappear when the center is (0,O). EXAMPLE 1 Find the circle that has a diameter from (1,7) to (5, 7). Solution The center is halfway at (3,7). So r = 2 and (x -3)2+ (y -7)2= 22. EXAMPLE 2 Find the center and radius of the circle x2 -6x + y2 -14y = -54.
Example 1. Consider the equation y2 = 4x + 12. Find the coordinates of the focus and the vertex and the equations of the directrix and the axis of symmetry. Graph the equation of the parabola. First, write the equation in the form (y – k)2 = 4p(x – h). y2 = 4x + 12 y2 = 4(x + 3) (y – 0)2 = 4(1)(x + 3) Factor. 4p = 4, so p = 1.
Find an equation for the parabolic shape of each cable. c. Complete the table by finding the heights 𝑦 of the suspension cables over the roadway at distances of 𝑥 meters from the center of the bridge
Example The mirrors in torches and car headlights are shaped like parabolas; microwave receivers on the roofs of buildings and satellite TV receivers also have parabolic shapes. Parabolas have the special property that radiation generated at a point, called the focus, is reflected in parallel rays off the parabola.
Writing Equations of Parabolas Date_____ Period____ Use the information provided to write the vertex form equation of each parabola. 1) Vertex at origin, Focus: (
