GRAPHING QUADRATIC FUNCTIONS WORKSHEET

Graph each quadratic function. 

1.  y = x²

2.  y = -x²

3.  y = 2x² - 8x + 6

4.  y = -½(x + 3)² + 4

5.  y = -(x + 2)(x - 4)

6.  Find the equation of the quadratic function with graph 

7.  Find the equation of the quadratic function with graph 

8.  Find the quadratic function which cuts the x-axis at 3 and -2 and which has y-intercept 24. Give your answer in the form y = ax² + bx + c.

Detailed Answer Key

1. Answer : 

The function is in standard form y = ax² + bx + c

a = 1, b = 0, and c = 0

Because a > 0, the parabola opens up.

Find and plot the vertex. The x-coordinate is :

x  =  -b/2a

Substitute. 

=  0/2(1)

=  0

The x-coordinate at the vertex is 0 and axis of symmetry is x = 0. 

The y-coordinate is :

y  =  0²

y  =  0

So, the vertex is (0, 0). 

Draw the axis of symmetry x = 0.

Plot two points on one side of the axis of symmetry, such as (-1, 1) and (-2, 2). Use symmetry to plot two more points, such as (1, 1) and (2, 2).

Draw a parabola through the plotted points.

2. Answer : 

The function is in standard form y = ax² + bx + c

a = -1, b = 0, and c = 0

Because a > 0, the parabola opens up.

Find and plot the vertex. The x-coordinate is :

x  =  -b/2a

Substitute. 

=  0/2(-1)

=  0

The x-coordinate at the vertex is 0 and axis of symmetry is x = 0. 

The y-coordinate is :

y  =  -0²

y  =  0

So, the vertex is (0, 0). 

Draw the axis of symmetry x = 0.

Plot two points on one side of the axis of symmetry, such as (-1, -1) and (-2, -2). Use symmetry to plot two more points, such as (1, -1) and (2, -2).

Draw a parabola through the plotted points.

3. Answer : 

The function is in standard form y = ax² + bx + c

a = 2, b = -8, and c = 6

Because a > 0, the parabola opens up.

Find and plot the vertex. The x-coordinate is :

x  =  -b/2a

Substitute. 

=  -(-8)/2(2)

=  8/4

=  2

The x-coordinate at the vertex is 2 and axis of symmetry is x = 2. 

The y-coordinate is :

y  =  2(2)² - 8(2) + 6

=  8 - 16 + 6

=  -2

So, the vertex is (2, -2). 

Draw the axis of symmetry x = 2.

Plot two points on one side of the axis of symmetry, such as (1, 0) and (0, 6). Use symmetry to plot two more points, such as (3, 0) and (4, 6).

Draw a parabola through the plotted points.

4. Answer : 

The function is in vertex form y = a(x - h)² + k.

a = -½, h = -3, and k = 4

Because a < 0, the parabola opens down.

To graph the function, first plot the vertex (h, k) = (-3, 4).

Draw the axis of symmetry x = -3 and plot two points on one side of it, such as (-1, 2) and (1, -4).

Use symmetry to complete the graph.

5. Answer : 

The quadratic function is in intercept form

y  =  a(x - p)(x - q)

where a = -1, p = -2, and q = 4.

The x-intercepts occur at (-2, 0) and (4, 0).

The axis of symmetry lies halfway between these points, at x = 1.

So, the x-coordinate of the vertex is x = 1 and the y-coordinate of the vertex is :

y  =  -(1 + 2)(1 - 4)

=  -(3)(-3)

=  9

The graph of the function is shown below.

6. Answer :

By observing the graph, we get that the vertex of the parabola is at (2, -20)

y = a(x - h)² + k

y = a(x - 2)² + (-20)

y = a(x - 2)² - 20

x-intercept of the parabola is 5.

Applying the point (5, 0), we get

0 = a(5 - 2)² - 20

20 = a(3)²

a = 20/9

y = (20/9)(x - 2)² - 20

7. Answer :

By observing the graph, the axis of symmetry is at x = 4, one of the x-intercept is at 7 and one of the y-intercept is at -2.

y = a(x - h)² + k

Here h = 4

y = a(x - 4)² + k

x-intercept is 7, then applying the point (7, 0), we get

0 = a(7 - 4)² + k

0 = 9a + k

9a + k = 0 ----(1)

y-intercept is -2, then applying the point (0, -2), we get

-2 = a(0 - 4)² + k

-2 = 16a + k

16a + k = -2 ----(2)

(1) - (2)

9a - 16a = -2

-7a = -2

a = 2/7

By applying a = 2/7 in (1), we get

9(2/7) + k = 0

k = -18/7

Applying the values of a and k, we get

y = (2/7)(x - 4)² + (-18/7)

So, the required equation is,

y = (2/7)(x - 4)² - (18/7)

8. Answer :

Find the quadratic function which cuts the x-axis at 3 and -2 and which has y-intercept 24. Give your answer in the form y = ax² + bx + c.

Since x -intercepts are 3 and -2, we write the equation in factored form.

y = a(x - p) (x - q)

Here p and q are 3 and -2

y = a(x - 3)(x + 2)

y-intercept = 24 when x = 0

24 = a(0 - 3)(0 + 2)

-6a = 24

a = -4

Applying the value of a, we get

y = -4(x - 3)(x + 2)

y = -4(x² + 2x - 3x - 6)

= -4(x² - 3x - 6)

y = -4x² + 12x + 24

So, the required equation is 

y = -4x² + 12x + 24

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