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 .

f (x) = 6 x^{2} - 5 \Leftrightarrow f (x) = 6 x^{2} + 0 x + (- 5)

We can see that

a = 6,

b = 0,

and

c = - 5 .

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

$a = 6$ , $b = 0$

x = - \frac{0}{2 ( 6 )}

▼

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 = f (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}, f (- \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.

f (x) = 6 x^{2} - 5

Substitute

$x = 0$

f (0) = 6 (0)^{2} - 5

▼

Simplify right-hand side

CalcPow

Calculate power

f (0) = 6 (0) - 5

Multiply

f (0) = 0 - 5

IdPropAdd

Identity Property of Addition

f (0) = - 5

We found the

y -

coordinate, and now we know that the vertex is

(0, - 5) .

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 $(0, - 5) .$ 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 = 1 .

f (x) = 6 x^{2} - 5

Substitute

$x = 1$

f (1) = 6 (1)^{2} - 5

▼

Simplify right-hand side

CalcPow

Calculate power

f (1) = 6 (1) - 5

Multiply

f (1) = 6 - 5

SubTerm

Subtract term

f (1) = 1

When

x = 1,

we have

y = 1 .

Therefore, the point

(1, 1)

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 $a = 6,$ which is positive, the parabola will open upwards. 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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