How to graph a parabola x=(y^2) – 4y + 3?

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If the question is to sketch the parabola, you plot the vertex and the ##x##- and ##y##-intercepts, and draw a smooth line through them.

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Warning! This is a long answer.

##x= y^2 -4y +3##

Note that ##x## is the dependent variable and ##y## is the independent variable.

We are going to get a sideways parabola.

Step 1. Define your variables.

The standard form for the equation of this parabola is

##x = ay^2 +by +c##

So

##a= 1##, ##b = -4##, and ##c = 3##

Step 2. Calculate and plot the vertex.

The vertex of the curve is given by

##y = -b/(2a) = -(-4)/(2×1) = -(-2) = 2##

Calculate the x-coordinate of the vertex.

##x= y^2 -4y +3 = (2)^2 -4(2) + 3 = 4 -8 + 3 = -1##

So the vertex is at (##-1,2##).

Plot your vertex point.

Step 3. Find the direction of the opening.

The parabola will be a sideways U opening either to the right (##⊂##) or to the left (##⊃##).

Since the coefficient ##a## is positive, the parabola opens in the positive direction (to the right).

Step 4. (optional) Draw the parabola’s axis of symmetry.

A parabola’s axis of symmetry is a line that runs through its middle and divides it in half.

For a quadratic of the form ##x = ay^2 +by +c##, the axis is a line that passes through the vertex and is parallel to the ##y## axis.

For our parabola, the axis is the line y = 2.

It’s not part of the parabola itself, but lightly marking this line on your graph can help you see how the parabola curves symmetrically.

Step 5. Calculate and plot the ##x##-intercept.

##x= y^2 -4y +3 = (0)^2 -4(0)+3 = 0 – 0 + 3 = 3##

The ##x##-intercept is at (##3,0##). Plot this point.

Step 6. Calculate and plot any ##y##-intercepts.

##f(y) = y^2 -4y +3 = 0##
##(y-3)(y-1) = 0##
##y-3 = 0## or ##y-1 = 0##
##y = 3## or ##y = 1##

The x-intercepts are at (##0,3##) and (##0,1##).

Add these points to the graph.

Step 7. Add any extra points to the graph.

The ##x##-intercept at (##3,0##) is 2 units below the axis. There should be a corresponding point 2 units above the axis at (##3,4##).

Plot this point.

Step 7. Draw a smooth parabola passing through all the points.



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