Quadratic Function Word Problems

 

1From the graph of the function f(x) = x2, graph the following translations:

1. y = x² + 2

2. y = x² − 2

3. y = (x + 2)²

4. y = (x + 2)²

5. y = (x − 2)² + 2

6. y = (x + 2)² − 2

2Find the vertex and determine the equation of the axis of symmetry for the following parabolas:

1. y = (x−1)² + 1

2. y = 3(x−1)² + 1

3. y = 2(x+1)² − 3

4. y = −3(x − 2)² − 5

5. y = x² − 7x −18

6. y = 3x² + 12x − 5

3Determine, without graphing, how many x-intersepts the following parabolas have:

1. y = x² − 5x + 3

2. y = 2x² − 5x + 4

3. y = x² − 2x + 4

4. y = −x² − x + 3

4Graph the following quadratic functions:

1. y = −x² + 4x − 3

2. y = x² + 2x + 1

3. y = x² + x + 1

5A quadratic function has an equation in the form y = x² + ax + a and passes through the point (1, 9). Calculate the value of a.

6The quadratic equation y = ax² + bx + c passes through the points (1,1), (0, 0) and (−1,1). Calculate the value of a, b and c.

7A parabola has its vertex at the point V(1, 1) and passes through the point (0, 2). Find its equation.


Solutions to the Quadratic Function Word Problems

1

From the graph of the function f(x) = x2, graph the following translations:

2. y = x² − 2

3. y = (x + 2)²

4. y = (x + 2)²

5. y = (x − 2)² + 2

6. y = (x + 2)² − 2

Parabola Translation

y = x²

Parabola TranslationParabola Translation

y = x² +2 y = x² −2

 

Parabola TranslationParabola Translation

y = (x + 2)²y = (x − 2)²

Parabola TranslationParabola Translation

y = (x − 2)² + 2 y = (x + 2)² − 2


2

Find the vertex and determine the equation of the axis of symmetry for the following parabolas:

1. y = (x − 1)² + 1

V = (1, 1)            x = 1

2. y = 3(x − 1)² + 1

V = (1, 1)            x = 1

3. y= 2(x + 1)² − 3

V = (−1, −3)            x = −1

4. y= −3(x − 2)² − 5

V = (2, −5)            x = 2

5. y = x² − 7x −18

V = (7/2, −121/ 4)            x = 7/2

6. y = 3x² + 12x − 5

V = (−2 , −17 )            x = −2


3

Determine, without graphing, how many x-intersepts the following parabolas have:

1. y = x² − 5x + 3

b² − 4ac = 25 − 12 > 0 two x-intercepts

2. y = 2x² − 5x + 4

b² − 4ac = 25 − 32 < 0 No x-intercept

3. y = x² − 2x + 4

b² − 4ac = 4 − 4 = 0 One x-intercept

4. y = −x² − x + 3

b² − 4ac = 1 + 12 > 0 Two x-intercepts


4

Graph the following quadratic functions:

1. y = −x² + 4x − 3

1. Vertex.

xv = − 4/ −2 = 2     yv = −2² + 4· 2 − 3 = 1        V(2, 1)

2. x-intercepts.

x² − 4x + 3 = 0

ecuación       (3, 0)      (1, 0)

3. y-intercept.

(0, −3)

Parabola


2. y = x² + 2x + 1

1. Vertex

x v = − 2/2 = −1     y v = (−1)² + 2 · (−1) + 1= 0        V(−1, 0)

2. x-intercepts.

x² + 2x + 1= 0

x-intercept Coincides with the point: (−1, 0)

3. y-intercept.

 (0, 1)

Parabola


3. y = x² + x + 1

1. Vertex.

xv = −1/ 2     yv = (−1/ 2)² + (−1/ 2) + 1= 3/4

V(−1/ 2, 3/ 4)

2. x-intercepts.

x² + x + 1= 0

1² − 4 < 0     No points.

3. y-intercept.

(0, 1)

Parabola

5

A quadratic function has an equation in the form y = x² + ax + a and passes through the point (1, 9). Calculate the value of a.

9 = 1² + a· 1 + a a = 4


6

The quadratic equation y = ax² + bx + c passes through the points (1,1), (0, 0) and (−1,1). Calculate the value of a, b and c.

1 = a · 1² + b · 1 + c

0 = a · 0² + b · 0 + c

1 = a · (−1)² + b · (−1) + c

a = 1 b = 0 c = 0


7

A parabola has its vertex at the point V (1, 1) and passes through the point (0, 2). Find its equation.

The value of the x coordinate of the vertex is 1.

1 = −b /2 a b = −2a

y = ax² + bx + c

f(0)=2

2 = c

f(1) = 1

1 = a + b + 2 1 = a −2a + 2

a = 1 b = −2

y = x2 − 2x + 2




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