Grafica funciones cuadráticas

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9-5 Solving Quadratic Equations by Graphing9-5 Solving Quadratic Equations

by Graphing

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Warm UpLesson PresentationLesson Quiz

Holt Algebra 1

9-5 Solving Quadratic Equations by Graphing

Warm Up

1. Grafica y = x2 + 4x + 3.

2. Identifica el vértice y los ceros de la función. vertex:(–2 , –1); zeros:–3, –1

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Resolver ecuaciones cuadráticas graficando.

Objetivo

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ecuación cuadráticaVocabulary

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Cada función cuadrática tiene está relacionada a una ecuación cuadrática. Una ecuación cuadrática es una ecuación que puede ser escrita en la forma ax2 + bx + c = 0, donde a, b, y c son números reales y a ≠ 0.

y = ax2 + bx + c0 = ax2 + bx + c

ax2 + bx + c = 0

Cuando se escribe una función cuadrática de una ecuación cuadrática se reemplaza y con 0. O sea que y = 0.

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Una manera de resolver una función cuadrática en forma estándar es graficando la función y encontrando los valores de x donde y = 0. En otras palabras, encontrando los ceros de la función. Hay que recordar que una función cuadrática puede tener dos, uno o ningún cero.

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Example 1A: Solving Quadratic Equations by Graphing Resuelve la ecuación graficando la función.

2x2 – 18 = 0 Paso 1 Escribe en forma de función.2x2 – 18 = y, or y = 2x2 + 0x – 18

Paso 2 Grafica la función.• El eje axis de simetría es x = 0.• El vértice es (0, –18). • Dos puntos adicionales son • (2, –10) y (3, 0)• Grafica los puntos y refléjalos a

través del eje de simetría.

(3, 0) ●x = 0

(2, –10) ●

(0, –18)●

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Example 1A Continued Resuelve la ecuación graficando la función.

Paso 3 Encuentra los ceros.2x2 – 18 = 0

Los ceros parecen ser 3 y –3.

Substituye 3 y –3 por x en la ecuación cuadrática. 0 =0

Verifica 2x2 – 18 =0 2(3)2 – 18 =0 2(9) – 18 =0 18 – 18 =0

2x2 – 18 = 0 2(–3)2 – 18 0

2(9) – 18 0 18 – 18 0

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Example 1B: Solving Quadratic Equations by Graphing Resuelve la ecuación graficando la función.

–12x + 18 = –2x2 Paso 1 Escribe en forma de función.

y = –2x2 + 12x – 18 Paso 2 Grafica la función.

• El eje de simetría es x = 3.• El vértice es (3, 0). • Dos puntos adicionales son (5, –8) y (4, –2).• Grafica los puntos y refléjalos a través del eje de simetría.

(5, –8)

(4, –2)

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●●

●

●

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x = 3(3, 0)

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Example 1B ContinuedResuelve la ecuación graficando la función.

Paso 3 Encuentra los ceros.El unico cero parece ser 3.

Verifica y = –2x2 + 12x – 180 –2(3)2 + 12(3) – 18 0 –18 + 36 – 18 0 0

You can also confirm the solution by using the Table function. Enter the function and press When y = 0, x = 3. The x-intercept is 3.

–12x + 18 = –2x2

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Example 1C: Solving Quadratic Equations by Graphing Solve the equation by graphing the related function.

2x2 + 4x = –3 Step 1 Write the related function.

y = 2x2 + 4x + 3 2x2 + 4x + 3 = 0

Step 2 Graph the function.Use a graphing calculator.Step 3 Find the zeros.The function appears to have no zeros.

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Example 1C: Solving Quadratic Equations by Graphing Solve the equation by graphing the related function.

2x2 + 4x = –3

The equation has no real-number solutions.Check reasonableness Use the table function.

There are no zeros in the Y1 column. Also, the signs of the values in this column do not change. The function appears to have no zeros.

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Check It Out! Example 1a Solve the equation by graphing the related function.

x2 – 8x – 16 = 2x2

Step 1 Write the related function. y = x2 + 8x + 16

Step 2 Graph the function.• The axis of symmetry is x = –4.• The vertex is (–4, 0). • The y-intercept is 16. • Two other points are (–3, 1) and (–2, 4).• Graph the points and reflect

them across the axis of symmetry.

x = –4

(–4, 0) ●

(–3, 1) ●

(–2 , 4) ●●

●

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Solve the equation by graphing the related function.

Check It Out! Example 1a Continued

Step 3 Find the zeros.The only zero appears to be –4.

Check y = x2 + 8x + 160 (–4)2 + 8(–4) + 16 0 16 – 32 + 16 0 0

x2 – 8x – 16 = 2x2

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6x + 10 = –x2 Step 1 Write the related function.y = x2 + 6x + 10

Check It Out! Example 1b

Step 2 Graph the function.• The axis of symmetry is x = –3 .• The vertex is (–3 , 1). • The y-intercept is 10. • Two other points (–1, 5) and (–2, 2)• Graph the points and reflect

them across the axis of symmetry.

x = –3

(–3, 1) ● (–2, 2) ●

(–1, 5) ●

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●

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6x + 10 = –x2

Check It Out! Example 1b Continued

Step 3 Find the zeros.There appears to be no zeros.

You can confirm the solution by using the Table function. Enter the function and press There are no negative terms in the Y1 table.

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–x2 + 4 = 0

Check It Out! Example 1c

Step 1 Write the related function.y = –x2 + 4

Step 2 Graph the function.Use a graphing calculator.Step 3 Find the zeros.

The function appears to have zeros at (2, 0) and (–2, 0).

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The equation has two real-number solutions.Check reasonableness Use the table function.

There are two zeros in the Y1 column. The function appears to have zeros at –2 and 2.

Check It Out! Example 1c Continued

–x2 + 4 = 0

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Example 2: Application A frog jumps straight up from the ground. The quadratic function f(t) = –16t2 + 12t models the frog’s height above the ground after t seconds. About how long is the frog in the air?

When the frog leaves the ground, its height is 0, and when the frog lands, its height is 0. So solve 0 = –16t2 + 12t to find the times when the frog leaves the ground and lands.Step 1 Write the related function

0 = –16t2 + 12ty = –16t2 + 12t

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Example 2 Continued

Step 2 Graph the function.Use a graphing calculator.

Step 3 Use to estimate the zeros.The zeros appear to be 0 and 0.75.The frog leaves the ground at 0 seconds and lands at 0.75 seconds.The frog is off the ground for about 0.75 seconds.

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Check 0 = –16t2 + 12t0 –16(0.75)2 + 12(0.75) 0 –16(0.5625) + 9 0 –9 + 9 0 0

Substitute 0.75 for t in the quadratic equation.

Example 2 Continued

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Check It Out! Example 2 What if…? A dolphin jumps out of the water. The quadratic function y = –16x2 + 32 x models the dolphin’s height above the water after x seconds. About how long is the dolphin out of the water? When the dolphin leaves the water, its height is 0, and when the dolphin reenters the water, its height is 0. So solve 0 = –16x2 + 32x to find the times when the dolphin leaves and reenters the water.

Step 1 Write the related function0 = –16x2 + 32xy = –16x2 + 32x

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Step 2 Graph the function.Use a graphing calculator.

Step 3 Use to estimate the zeros.The zeros appear to be 0 and 2.The dolphin leaves the water at 0 seconds and reenters at 2 seconds.The dolphin is out of the water for about 2 seconds.

Check It Out! Example 2 Continued

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Check It Out! Example 2 Continued

Check 0 = –16x2 + 32x0 –16(2)2 + 32(2) 0 –16(4) + 64 0 –64 + 64 0 0

Substitute 2 for x in the quadratic equation.

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Lesson Quiz

Resuelve la ecuación graficando la función.1. 3x2 – 12 = 02. x2 + 2x = 83. 3x – 5 = x2

4. 3x2 + 3 = 6x5. A rocket is shot straight up from the ground.

The quadratic function f(t) = –16t2 + 96t models the rocket’s height above the ground after t seconds. How long does it take for the rocket to return to the ground?

2, –2 –4, 2 no tiene solución1

6 s

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