Question

# Find the y-intercept of the following function

f(x) = 0.3x^{2} + 0.6x – 0.7

Hint:

### For a quadratic function is in standard form, f(x)=ax2+bx+c.

The *y*-intercept is the point where a graph crosses the *y*-axis. In other words, it is the value of *y* when x=0.

## The correct answer is: -0.7.

### We have given a function

f(x) = 0.3x^{2} + 0.6x – 0.7

We will compare the given equation with the standard equation f(x)=ax^{2}+bx+c.

We know that for y intercept , x = 0

So, for finding y- intercept

f(x) = y = 0.3(0)^{2} + 0.6(0) – 0.7

y = -0.7

On comparing with the standard form y-intercept is equal to c

y-intercept of given quadratic function is -0.7.

### Related Questions to study

### Find the y-intercept of the following function

f(x) = 2x^{2} – 4x – 6

f(x) = 2x

^{2}– 4x – 6

We will compare the given equation with the standard equation f(x)=ax

^{2}+bx+c.

We know that for y intercept , x = 0

So, for finding y- intercept

f(x) = y = 2(0)

^{2 }– 4(0) – 6

y = -6

On comparing with the standard form y-intercept is equal to c

y-intercept of given quadratic function is -6.

### Find the y-intercept of the following function

f(x) = 2x^{2} – 4x – 6

f(x) = 2x

^{2}– 4x – 6

We will compare the given equation with the standard equation f(x)=ax

^{2}+bx+c.

We know that for y intercept , x = 0

So, for finding y- intercept

f(x) = y = 2(0)

^{2 }– 4(0) – 6

y = -6

On comparing with the standard form y-intercept is equal to c

y-intercept of given quadratic function is -6.

### Points (2, -1), (-2, 7), (1, -2), (0, -1) and (4, 7) lie on graph of a quadratic function

1.Find axis of symmetry of graph

2.Find the vertex

3.Find the y-intercept

4.Over what interval does the function increase

We can see that the axis of symmetry passes through x= 1

Therefore, the vertex point is (1,-2)

Y intercept is the point at which x = 0 which is (0,-1)

y- intercept= -1

### Points (2, -1), (-2, 7), (1, -2), (0, -1) and (4, 7) lie on graph of a quadratic function

1.Find axis of symmetry of graph

2.Find the vertex

3.Find the y-intercept

4.Over what interval does the function increase

We can see that the axis of symmetry passes through x= 1

Therefore, the vertex point is (1,-2)

Y intercept is the point at which x = 0 which is (0,-1)

y- intercept= -1

### Estimate the coordinates of the vertex of the graph of f(x) = 1.25x^{2} -2x -1 below. Then explain how to find the exact coordinates

f(x) = 1.25x

^{2}-2x -1

This quadratic function is in standard form, f(x)=ax

^{2}+bx+c.

For every quadratic function in standard form the axis of symmetry is given by the formula x=−b/2a.

In f(x)= 1.25x

^{2}-2x -1, a= 1.25, b= -2, and c= -1. So, the equation for the axis of symmetry is given by

x = −(-2)/2(1.25)

x = 2/2.5

x = 0.8

The equation of the axis of symmetry for f(x)= 1.25x

^{2}-2x -1 is x = 0.8.

The x coordinate of the vertex is the same:

h = 0.8

The y coordinate of the vertex is :

k = f(h)

k = 1.25h

^{2}-2h -1

k = 1.25(0.8)

^{2}- 2(0.8) - 1

k = 0.8 - 1.6 – 1

k = -1.8

Therefore, the vertex is (0.8 , -1.8)

### Estimate the coordinates of the vertex of the graph of f(x) = 1.25x^{2} -2x -1 below. Then explain how to find the exact coordinates

f(x) = 1.25x

^{2}-2x -1

This quadratic function is in standard form, f(x)=ax

^{2}+bx+c.

For every quadratic function in standard form the axis of symmetry is given by the formula x=−b/2a.

In f(x)= 1.25x

^{2}-2x -1, a= 1.25, b= -2, and c= -1. So, the equation for the axis of symmetry is given by

x = −(-2)/2(1.25)

x = 2/2.5

x = 0.8

The equation of the axis of symmetry for f(x)= 1.25x

^{2}-2x -1 is x = 0.8.

The x coordinate of the vertex is the same:

h = 0.8

The y coordinate of the vertex is :

k = f(h)

k = 1.25h

^{2}-2h -1

k = 1.25(0.8)

^{2}- 2(0.8) - 1

k = 0.8 - 1.6 – 1

k = -1.8

Therefore, the vertex is (0.8 , -1.8)

### To identify the y-intercept of quadratic function, would you choose to use vertex form or standard form? Explain

If we see the standard form, we get an constant number c on putting the value of x=0. This value of c is called as y-intercept.

But in vertex form it is not so easy to get y-intercept because in this form the constant term is not distinct as in the standard form

So, Standard form is the right choice for identifying the y-intercept of quadratic function.

### To identify the y-intercept of quadratic function, would you choose to use vertex form or standard form? Explain

If we see the standard form, we get an constant number c on putting the value of x=0. This value of c is called as y-intercept.

But in vertex form it is not so easy to get y-intercept because in this form the constant term is not distinct as in the standard form

So, Standard form is the right choice for identifying the y-intercept of quadratic function.

### A water balloon is tossed into the air. The function h(x) = -0.5(x-4)^{2} + 9 gives the height, in feet, of the balloon from the surface of a pool as a function of the balloon’s horizontal distance from where it was first tossed. Will the balloon hit the ceiling 12 ft above the pool? Explain

h(x) = -0.5(x-4)

^{2}+ 9

The given equation is in the vertex form

We know vertex is the point at maximum height .

And y coordinate of vertex is the maximum height of the ball

On comparing with the vertex form we get that

h = 4

k = 9

Therefore, here vertex is (4, 9)

Here, we have y-coordinate as 9

So, the maximum height of ball is 9

We have given the ceiling height as 12 ft

So the height of ceiling is more than maximum height of ball , so ball will not hit the ceiling.

### A water balloon is tossed into the air. The function h(x) = -0.5(x-4)^{2} + 9 gives the height, in feet, of the balloon from the surface of a pool as a function of the balloon’s horizontal distance from where it was first tossed. Will the balloon hit the ceiling 12 ft above the pool? Explain

h(x) = -0.5(x-4)

^{2}+ 9

The given equation is in the vertex form

We know vertex is the point at maximum height .

And y coordinate of vertex is the maximum height of the ball

On comparing with the vertex form we get that

h = 4

k = 9

Therefore, here vertex is (4, 9)

Here, we have y-coordinate as 9

So, the maximum height of ball is 9

We have given the ceiling height as 12 ft

So the height of ceiling is more than maximum height of ball , so ball will not hit the ceiling.

### Sage began graphing f(x) = -2x^{2} + 4x + 9 by finding the axis of symmetry x = -1. Explain the error Sage made?

^{2}+ 4x + 9

We will first find the axis of symmetry ‘

x = -b/ 2a

x = -4 / 2 (-2)

x = -4/ -4

x = 1

This detects the error made by sage that he finds axis of symmetry as x = -1

### Sage began graphing f(x) = -2x^{2} + 4x + 9 by finding the axis of symmetry x = -1. Explain the error Sage made?

^{2}+ 4x + 9

We will first find the axis of symmetry ‘

x = -b/ 2a

x = -4 / 2 (-2)

x = -4/ -4

x = 1

This detects the error made by sage that he finds axis of symmetry as x = -1

### How are the form and graph of f(x) = ax^{2} + bx + c similar to the form and graph of g(x) = ax^{2} + bx? How are they different?

f(x) = ax

^{2}+ bx + c

g(x) = ax

^{2}+ bx

By analyzing both the equations we get that both the equation are of same format

In first equation the constant c is present whereas in second equation constant c is zero. So we know that in the standard form this constant represents the y- intercept of the curve.

So the first curve intercepts y axis at (0,c)

And second curve intercepts y axis at (0,0)

### How are the form and graph of f(x) = ax^{2} + bx + c similar to the form and graph of g(x) = ax^{2} + bx? How are they different?

f(x) = ax

^{2}+ bx + c

g(x) = ax

^{2}+ bx

By analyzing both the equations we get that both the equation are of same format

In first equation the constant c is present whereas in second equation constant c is zero. So we know that in the standard form this constant represents the y- intercept of the curve.

So the first curve intercepts y axis at (0,c)

And second curve intercepts y axis at (0,0)

### How is the standard form of a quadratic function different from vertex form?

In both forms, y is the y-coordinate, x is the x-coordinate, and a is the constant that tells you whether the parabola is facing up (+a) or down (−a).It is as if the parabola was a bowl of applesauce; if there's a +a, I can add applesauce to the bowl; if there's a −a, I can shake the applesauce out of the bowl.

The difference between a parabola's standard form and vertex form is that the vertex form of the equation also gives you the parabola's vertex: (h, k).

### How is the standard form of a quadratic function different from vertex form?

In both forms, y is the y-coordinate, x is the x-coordinate, and a is the constant that tells you whether the parabola is facing up (+a) or down (−a).It is as if the parabola was a bowl of applesauce; if there's a +a, I can add applesauce to the bowl; if there's a −a, I can shake the applesauce out of the bowl.

The difference between a parabola's standard form and vertex form is that the vertex form of the equation also gives you the parabola's vertex: (h, k).

### Suppose the path of the ball in above figure is f(x) = -0.25(x-1)^{2} + 6.25. Find the ball’s initial and maximum heights.

^{2}+ 6.25

This is in the vertex form

Therefore on comparing with the vertex form of a quadratic equation is y=a(x−h)

^{2}+k. we get, h = 1 , k = 6.25

Therefore, vertex is (h,k) = (1, 6.25)

So the maximum height of the ball is 6.25 .

Converting the equation in standard form

f(x) = -0.25(x-1)^{2} + 6.25

f(x) = -0.25(x^{2}+1 – 2x) + 6.25

f(x) = -0.25x^{2} – 0.25 + 0.5x + 6.25

f(x) = -0.25x^{2} +0.5 x + 6.00

Therefore, y intercept is c = 6

Which is the initial height of the ball i.e 6

### Suppose the path of the ball in above figure is f(x) = -0.25(x-1)^{2} + 6.25. Find the ball’s initial and maximum heights.

^{2}+ 6.25

This is in the vertex form

Therefore on comparing with the vertex form of a quadratic equation is y=a(x−h)

^{2}+k. we get, h = 1 , k = 6.25

Therefore, vertex is (h,k) = (1, 6.25)

So the maximum height of the ball is 6.25 .

Converting the equation in standard form

f(x) = -0.25(x-1)^{2} + 6.25

f(x) = -0.25(x^{2}+1 – 2x) + 6.25

f(x) = -0.25x^{2} – 0.25 + 0.5x + 6.25

f(x) = -0.25x^{2} +0.5 x + 6.00

Therefore, y intercept is c = 6

Which is the initial height of the ball i.e 6

### Mia tosses a a ball to her dog. The function -0.5(x-2)^{2} + 8 represents the ball path.

Solution:- The standard quadratic form is ax

^{2}+bx+c=y, the vertex form of a quadratic equation is y=a(x−h)

^{2}+k.

In both forms, y is the y-coordinate, x is the x-coordinate, and a is the constant that tells you whether the parabola is facing up (+a) or down (−a).

The vertex form of the equation also gives you the parabola's vertex: (h,k).

This vertex is the highest point on the trajectory of the ball tossed by mia.

2.What does the standard form of a function tell you about the situation?

Solution:- The standard quadratic form is ax

^{2}+bx+c=y

In this the term c gives us the y-intercept of the curve

In the given case it gives the height from which the ball is thrown.

-0.5(x-2)^{2} + 8 = y

- 0.5( x^{2}+ 4 – 4x) + 8 = y

-0.5x^{2} – 2 + 2x + 8 = y

-0.5x^{2} + 2x + 6 = y

C = 6

Which is height from which the ball is thrown.

### Mia tosses a a ball to her dog. The function -0.5(x-2)^{2} + 8 represents the ball path.

Solution:- The standard quadratic form is ax

^{2}+bx+c=y, the vertex form of a quadratic equation is y=a(x−h)

^{2}+k.

In both forms, y is the y-coordinate, x is the x-coordinate, and a is the constant that tells you whether the parabola is facing up (+a) or down (−a).

The vertex form of the equation also gives you the parabola's vertex: (h,k).

This vertex is the highest point on the trajectory of the ball tossed by mia.

2.What does the standard form of a function tell you about the situation?

Solution:- The standard quadratic form is ax

^{2}+bx+c=y

In this the term c gives us the y-intercept of the curve

In the given case it gives the height from which the ball is thrown.

-0.5(x-2)^{2} + 8 = y

- 0.5( x^{2}+ 4 – 4x) + 8 = y

-0.5x^{2} – 2 + 2x + 8 = y

-0.5x^{2} + 2x + 6 = y

C = 6

Which is height from which the ball is thrown.