Question

The equation of the curve is

y=x ^2+ax+b where
a and b are integers. The points
(0,−5) and (5,0) lie on the curve. Find the coordinates of the turning point of the curve.

Answer 1

The equation of the curve is y = x^2 −4x −5, and the coordinates of the turning point are (2, −4), found by applying the vertex formula for a parabola and using the given points to solve for coefficients a and b.

Step-by-step explanation:

To find the equation of the curve “y = x^2 + ax + b” that passes through the points (0,−5) and (5,0), we can use the given points to solve for 'a' and 'b'.

Substituting (0,−5) into the equation, we get:

• −5 = 0^2 + a∗0 + b
• b = −5

Substituting (5,0) into the equation, we get:

• 0 = 5^2 + 5a −5
• 25 + 5a −5 = 0
• 5a = 0 −20
• a = −4

Thus, the equation of the curve is y = x^2 −4x −5.

To find the coordinates of the turning point of the curve, we need to complete the square or use the fact that for a quadratic equation y = ax^2 + bx + c, the turning point, also known as the vertex, is at x = −b/(2a). In this case, a = 1 and b = −4, so the x-coordinate of the turning point is x = −(−4)/(2∗1) = 2. Substituting back into the equation, we get the y-coordinate as y = 2^2 −4∗2 −5 = 4 −8 −5 = −4. Therefore, the turning point is at (2, −4).