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QuestionFind the coordinates of the minimum point of the graph of x2−7x+11x^{2}-7x+11.

Studdy Solution

STEP 1

Assumptions1. The given function is f(x)=x−7x+11f(x) = x^{}-7 x+11 . The function is a quadratic function, and its graph is a parabola.
3. The parabola opens upwards because the coefficient of x^ is positive.
4. The minimum point of the parabola is the vertex of the parabola.

STEP 2

The formula to find the x-coordinate of the vertex of a parabola given by f(x)=ax2+bx+cf(x) = ax^2 + bx + c is h=−b2ah = -\frac{b}{2a}.

STEP 3

In our function, a=1a =1 and b=−7b = -7. Plug these values into the formula to find the x-coordinate of the vertex.
h=−−72⋅1h = -\frac{-7}{2 \cdot1}

STEP 4

Calculate the x-coordinate of the vertex.
h=−−72=3.h = -\frac{-7}{2} =3.

STEP 5

To find the y-coordinate of the vertex, we substitute hh into the function f(x)f(x).
k=f(3.5)=(3.5)2−7⋅3.5+11k = f(3.5) = (3.5)^{2}-7 \cdot3.5+11

STEP 6

Calculate the y-coordinate of the vertex.
k=(3.5)2−⋅3.5+11=12.25−24.5+11=−1k = (3.5)^{2}- \cdot3.5+11 =12.25 -24.5 +11 = -1The coordinates of the minimum point of the graph of x2−x+11x^{2}- x+11 are (3.5,−1)(3.5, -1).

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