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Graphing Parabolas and Identifying Focus, Directrix, and Axis of Symmetry

Learn how to graph equations of parabolas and identify the focus, directrix, and axis of symmetry. Understand the standard form and use it to write equations.

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Graphing Parabolas and Identifying Focus, Directrix, and Axis of Symmetry

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  1. 18 18 STEP 1 Rewrite the equation in standard form. x = – EXAMPLE 1 Graph an equation of a parabola Graphx= – y2. Identify the focus, directrix, and axis of symmetry. SOLUTION Write original equation. Multiply each side by–8. – 8x = y2

  2. STEP 2 STEP 3 Identify the focus, directrix, and axis of symmetry. The equation has the form y2 = 4pxwhere p = – 2. The focus is (p, 0), or (– 2, 0). The directrix is x = – p, or x = 2. Because yis squared, the axis of symmetry is the x - axis. Draw the parabola by making a table of values and plotting points. Because p < 0, the parabola opens to the left. So, use only negative x - values. EXAMPLE 1 Graph an equation of a parabola

  3. EXAMPLE 1 Graph an equation of a parabola

  4. Write an equation of the parabola shown. 3 2 3 2 The graph shows that the vertex is (0, 0) and the directrix is y = – p = for pin the standard form of the equation of a parabola. – ( )y 32 x2= 4 Substitute for p EXAMPLE 2 Write an equation of a parabola SOLUTION x2 = 4py Standard form, vertical axis of symmetry x2 = 6y Simplify.

  5. Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola. 1. y2 = –6x STEP 1 Rewrite the equation in standard form. 3 2 y2 = 4(– )x for Examples 1, and 2 GUIDED PRACTICE SOLUTION

  6. STEP 2 Identify the focus, directrix, and axis of symmetry. The equation has the form y2 = 4pxwhere p = – . The focus is (p, 0), or (– , 0). The directrix is x = – p, or x = . Because yis squared, the axis of symmetry is the x - axis. 3 3 3 2 2 2 for Examples 1, and 2 GUIDED PRACTICE

  7. STEP 3 Draw the parabola by making a table of values and plotting points. Because p < 0, the parabola opens to the left. So, use only negative x - values. 2.45 4.24 4.90 5.48 2.45 for Examples 1, and 2 GUIDED PRACTICE

  8. Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola. 2. x2 = 2y for Examples 1 and 2 GUIDED PRACTICE SOLUTION

  9. 14 14 STEP 1 y = – x2 Rewrite the equation in standard form. for Examples 1 and 2 GUIDED PRACTICE Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola. 3. y = – x2 SOLUTION Write original equation. – 4y = x2 Multiply each side by – 4.

  10. STEP 2 equation for Examples 1 and 2 GUIDED PRACTICE directrix axis of symmetry focus 0, –1 y = 1 Verticalx = 0 x2 = – 4

  11. STEP 3 Draw the parabola by making a table of values and plotting points. Because p < 0, the parabola opens to the left. So, use only negative y - values. y x 4 2 2.83 3.46 4.47 for Examples 1 and 2 GUIDED PRACTICE

  12. Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola. 4. x = – y2 13 STEP 1 Rewrite the equation in standard form. 13 y2 x = – for Examples 1 and 2 GUIDED PRACTICE SOLUTION Write original equation. 3x = y2 Multiply each side by 3.

  13. STEP 2 equation 0, x = – y2 = 4 x 3 3 3 4 4 4 for Examples 1 and 2 GUIDED PRACTICE directrix axis of symmetry focus Horizontaly = 0

  14. STEP 3 Draw the parabola by making a table of values and plotting points. Because p < 0, the parabola opens to the left. So, use only negative x - values. y x 3.46 1.73 2.45 3 3.87 for Examples 1 and 2 GUIDED PRACTICE

  15. for Examples 1 and 2 GUIDED PRACTICE Write the standard form of the equation of the parabola with vertex at (0, 0) and the given directrix or focus. 5. Directrix: y = 2 SOLUTION x2 = 4py Standard form, vertical axis of symmetry x2 =4 (–2)y Substitute –2 for p x2 = – 8y Simplify.

  16. for Examples 1 and 2 GUIDED PRACTICE Write the standard form of the equation of the parabola with vertex at (0, 0) and the given directrix or focus. 6. Directrix: x = 4 SOLUTION y2 = 4px Standard form, vertical axis of symmetry y2= 4 (–4)x Substitute –4 for p y2 = – 16x Simplify.

  17. for Examples 1 and 2 GUIDED PRACTICE Write the standard form of the equation of the parabola with vertex at (0, 0) and the given directrix or focus. 7. Focus: (–2, 0) SOLUTION y2 = 4px Standard form, vertical axis of symmetry y2= 4 (–2)x Substitute –2 for p y2 = – 8x Simplify.

  18. for Examples 1 and 2 GUIDED PRACTICE Write the standard form of the equation of the parabola with vertex at (0, 0) and the given directrix or focus. 8. Focus: (0, 3) SOLUTION x2 = 4py Standard form, vertical axis of symmetry x2= 4 (3)y Substitute 3 for p x2 = 12y Simplify.

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