Question

Melanie draws the graph of y=x^2+ax+b

Answer 1

Answers:

point A is located at (3, 0)

The turning point is located at (-2, -25)

Explanation

Plug in x = 0 to have y = x^2+ax+b turn into y = b.

Since the y intercept is -21, this must mean b = -21.

The equation is now y = x^2+ax-21.

Plug in x = -7 and y = 0 to solve for 'a'. After doing so, you should get a = 4.

The equation is now y = x^2+4x-21 and this factors to y = (x+7)(x-3)

Solving (x+7)(x-3) = 0 leads to x = -7 and x = 3.

The roots or x intercepts are -7 and 3. Therefore, point A is located at (3,0)

The x coordinate of the turning point occurs at the midpoint of the x intercepts. This is due to the parabola's symmetry.

Add up the roots and divide in half: (-7+3)/2 = -4/2 = -2

The turning point happens when x = -2. This is the axis of symmetry.

Plug this into the equation we found earlier to get the y coordinate of the turning point.

y = x^2+4x-21

y = (-2)^2+4(-2)-21

y = -25

The turning point is located at (-2,-25)

For any parabola, the turning point is the vertex. In this case, the vertex is the lowest point.

Extra info: The vertex form of the equation is y = (x+2)^2 - 25

Answer 2

There ya go

Explanation:

To find the coordinates of point A, we know that the graph crosses the x-axis at (-7,0) and point A. Since the x-coordinate of point A is unknown, let's call it (x,0).

Since the graph crosses the y-axis at (0, -21), we can substitute these coordinates into the equation y = x² + ax + b to solve for a and b.

Substituting (0, -21) into the equation, we have:

-21 = 0² + a(0) + b

-21 = b

So now we have b = -21.

Next, we substitute (-7,0) into the equation:

0 = (-7)² + a(-7) + (-21)

0 = 49 - 7a - 21

0 = 28 - 7a

Simplifying further, we have:

7a = 28

a = 4

Now that we have the values of a and b, we can determine the coordinates of point A.

Substituting a = 4 and b = -21 into the equation, we have:

y = x² + 4x - 21

To find the x-coordinate of point A, we set y = 0:

0 = x² + 4x - 21

Factoring or using the quadratic formula, we find that the x-coordinate of point A is 3.

Therefore, the coordinates of point A are (3, 0).

To find the coordinates of the turning point, we can use the formula x = -b/2a. Plugging in a = 4 and b = -21, we have:

x = -4 / (2 * 4)

x = -1

Now, substitute x = -1 into the equation y = x² + 4x - 21 to find the y-coordinate of the turning point:

y = (-1)² + 4(-1) - 21

y = 1 - 4 - 21

y = -24

Therefore, the coordinates of the turning point are (-1, -24).

In summary:

(a) The coordinates of point A are (3, 0).

(b) The coordinates of the turning point are (-1, -24).