To draw the graph of the given quadratic function, written in standard form, we must start by identifying the values of

a,

b,

and

c .

g (x) = - x^{2} - 1 \Leftrightarrow g (x) = - 1 x^{2} + 0 x + (- 1)

We can see that

a = - 1,

b = 0,

and

c = - 1 .

Now, we will follow four steps to graph the function.

Find the axis of symmetry.
Calculate the vertex.
Identify the $y -$ intercept and its reflection across the axis of symmetry.
Connect the points with a parabola.

Finding the Axis of Symmetry

The axis of symmetry is a vertical line with equation

x = - \frac{b}{2 a} .

Since we already know the values of

a

and

b,

we can substitute them into the formula.

x = - \frac{b}{2 a}

SubstituteII

$a = - 1$ , $b = 0$

x = - \frac{0}{2 ( - 1 )}

▼

Simplify right-hand side

ZeroPropMult

Zero Property of Multiplication

x = 0

The axis of symmetry of the parabola is the vertical line with equation

x = 0 .

Calculating the Vertex

To calculate the vertex, we need to think of

y

as a function of

x,

y = g (x) .

We can write the expression for the vertex by stating the

x -

and

y -

coordinates in terms of

a

and

b .

Vertex : (- \frac{b}{2 a}, g (- \frac{b}{2 a}))

Note that the formula for the

x -

coordinate is the same as the formula for the axis of symmetry, which is

x = 0 .

Thus, the

x -

coordinate of the vertex is also

0 .

To find the

y -

coordinate, we need to substitute

0

for

x

in the given equation.

g (x) = - x^{2} - 1

Substitute

$x = 0$

g (0) = - (0)^{2} - 1

▼

Simplify right-hand side

CalcPow

Calculate power

g (0) = 0 - 1

IdPropAdd

Identity Property of Addition

g (0) = - 1

We found the

y -

coordinate, and now we know that the vertex is

(0, - 1) .

Identifying the $y -$ intercept and its Reflection

The $y -$ intercept of the graph of a quadratic function written in standard form is given by the value of $c .$ Thus, the point where our graph intercepts the $y -$ axis is $(0, - 1) .$ Let's plot this point.

We can see that the

y -

intercept lies on the axis of symmetry, which means we cannot find another point when the

y -

intercept is reflected.For this reason, we will now plot a point on the curve by choosing an

x -

value and calculating its corresponding

y -

value. Let's try

x = 2 .

g (x) = - x^{2} - 1

Substitute

$x = 2$

g (2) = - (2)^{2} - 1

▼

Simplify right-hand side

CalcPow

Calculate power

g (2) = - (4) - 1

Distr

Distribute $- 1$

g (2) = - 4 - 1

SubTerms

Subtract terms

g (2) = - 5

When

x = 2,

we have

y = - 5 .

Thus, the point

(2, - 5)

lies on the curve. Let's plot this point and reflect it across the axis of symmetry.

Connecting the Points

We can now draw the graph of the function. Since $a = - 1,$ which is negative, the parabola will open downwards. Let's connect the three points with a smooth curve.

Finding the Axis of Symmetry

Calculating the Vertex

Identifying the y-intercept and its Reflection

Connecting the Points

Identifying the $y -$ intercept and its Reflection

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