Question

Y=-2(x-1)^2+8 a) Determine the coordinates of its turning point and state whether it is a Maximum or minimum B) Determine the y-intercept C)Determine the coordinates of the x-intercepts(if any)

Answer 1

(a) The coordinates of the inflection point are (1; 8) and, being the value of a less than zero, then these coordinates indicate the maximum of the function.

(b) The y-intercept has a value of 6

(c) The coordinates of the x-intercepts have a value of x=3 and x=-1

Explanation:

You know y= -2*(x-1)² + 8

(a) Every quadratic function can be expressed by the canonical form:

f(x)=y= a*( x- h)² + k

where a is the principal coefficient and the ordered pair (h; k) are the coordinates of the vertex of the parabola.

The quadratic function will have a maximum or minimum value at the vertex. If a> 0, the parabola is concave, that is, it will have its branches upwards, therefore the vertex will indicate the minimum of the function. But if a <0 the parabola is convex and the vertex will indicate the maximum of the function.

Comparing the given equation with the canonical form of a quadratic function you can see that:

• a=-2
• h=1
• k=8

Then, the coordinates of the inflection point are (1; 8) and, being the value of a less than zero, then these coordinates indicate the maximum of the function.

(b) The y-intercept arises from evaluating the function at x = 0. So, in this case:

y= -2*(0-1)² + 8

Solving:

y= -2*(-1)² + 8

y= -2*1 + 8

y= -2+8

y= 6

The y-intercept has a value of 6

(c) The roots or coordinates of the x-intercepts arise from evaluating the function at y = 0. So, in this case:

0= -2*(x-1)² + 8

Solving:

-8= -2*(x-1)²

`(-8)/(-2) =(x-1)^(2)`

4= (x-1)²

√4= |x-1|

⇒ 2=x-1 → 2+1=x → 3=x

⇒ -2=x-1 → -2+1=x → -1=x

The coordinates of the x-intercepts have a value of x=3 and x=-1