Question

The axis of symmetry for the function f(x) = −x2 − 10x + 16 is x = −5. What are the coordinates of the vertex of the graph?

(−5, 41)
(−5, 56)
(−5, 76)
(−5, 91)

Answer 1

we know that

The equation of a vertical parabola in vertex form is equal to

$y=a(x-h)^{2} +k$

where

(h,k) is the vertex

if $a > 0$ -----> the parabola open upward (vertex is a minimum)

if $a < 0$ -----> the parabola open downward (vertex is a maximum)

In a vertical parabola the axis of symmetry is equal to

$x=h$

In this problem we have

$f(x)=-x^{2}-10x+16$

Complete the square

$f(x)-16=-(x^{2}+10x)$

$f(x)-16-25=-(x^{2}+10x+25)$

$f(x)-41=-(x+5)^{2}$

$f(x)=-(x+5)^{2}+41$

The parabola open downward

the vertex is the point $(-5,41)$ (is a maximum)

therefore

the answer is

$(-5,41)$

Answer 2

Considering the definition of axis of symmetry and vertex of a quadratic function, the correct answer is the first option: the coordinates of the vertex of the graph is (-5,41).

Quadratic function

The general form of a quadratic function is f(x  = ax²+ bx+ c, whose graph is a parabola.

Axis of symmetry and vertex of a quadratic function

Quadratic functions have a maximum (if a<0) or a minimum (if a>0). This point is the vertex of the parabola.

That is, the vertex of a quadratic equation or parabola is the highest or lowest point on the graph corresponding to that function. The vertex is in the plane of symmetry of the parabola; anything that happens to the left of this point will be an exact reflection of what happens to the right.

In other words, the vertex divides the graph into two halves that are mirror images of each other, so that the axis of symmetry always passes through the vertex.

Vertex of f(x) = −x² − 10x + 16

In this case, you know that the axis of symmetry for the function is x = −5.

So the vertex on the x-axis has a value of -5. To calculate the value of the vertex on the y-axis, you must substitute the value of the vertex on the x-axis in the function:

f(-5) = −(-5)² − 10×(-5) + 16

Solving:

f(-5)= 41

In summary, the correct answer is the first option: the coordinates of the vertex of the graph is (-5,41).

Learn more about the vertex of a quadratic function:

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