To graph the quadratic function given in intercept form

f (x) = a (x - p) (x - q),

we must start by identifying the values of

a,

p,

and

q .

f (x) = 2 (x - 5) (x - 1) ⇕ f (x) = 2 (x - 5) (x - 1)

We can see that

a = 2,

p = 5,

and

q = 1 .

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

Identify the $x -$ intercepts.
Plot the $x -$ intercepts and find the axis of symmetry.
Calculate the vertex and plot it.
Connect the points with a parabola.
Identifying the $x -$ intercepts

Recall that the $x -$ intercepts of a function written in intercept form are the values of $p$ and $q .$ Thus, the points where our function intercepts the $x -$ axis are $(5, 0)$ and $(1, 0) .$

Finding the Axis of Symmetry
The axis of symmetry is halfway between the $x -$ intercepts, $(p, 0)$ and $(q, 0) .$ Therefore, it has the following equation.
$x = \frac{p + q}{2}$
Since we already know the values of $p$ and $q,$ we can substitute them into the formula.
$x = \frac{p + q}{2}$
SubstituteII
$p = 5$ , $q = 1$
$x = \frac{5 + 1}{2}$

▼

Simplify right-hand side

AddTerms
Add terms
$x = \frac{6}{2}$
CalcQuot
Calculate quotient

$x = 3$
The axis of symmetry of the parabola is the vertical line with equation $x = 3 .$ Let's plot the $x -$ intercepts and the axis of symmetry on a coordinate plane.

Calculating the Vertex
The $x -$ coordinate of the vertex is the same as the formula for the axis of symmetry, $x = 3 .$ To find the $y -$ coordinate, we need to substitute $3$ for $x$ in the given equation.
$f (x) = 2 (x - 5) (x - 1)$
Substitute
$x = 3$
$f (3) = 2 (3 - 5) (3 - 1)$

▼

Simplify right-hand side

SubTerms
Subtract terms
$f (3) = 2 (- 2) (2)$
Multiply
Multiply

$f (3) = - 8$
We found the $y -$ coordinate, and now we know that the vertex is $(3, - 8) .$ Let's plot it on our coordinate plane.

Connecting the Points

We can now draw the graph of the function. Since $a = 2,$ which is positive, the parabola will open upwards. Let's connect the three points with a smooth curve.

Identifying the x-intercepts

Finding the Axis of Symmetry

Calculating the Vertex

Connecting the Points

Identifying the $x -$ intercepts

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