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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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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
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
EXAMPLE 1 Graph an equation of a parabola
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.
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
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
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
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
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.
STEP 2 equation for Examples 1 and 2 GUIDED PRACTICE directrix axis of symmetry focus 0, –1 y = 1 Verticalx = 0 x2 = – 4
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
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.
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
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
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.
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.
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.
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.