2.4.11 Graphing parabolas

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2. Quantitative reasoning

2.4. Geometry

Graphing parabolas

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You’ve probably seen many problems that include equations like this:

$y = a x^{2} + b x + c$

You may also be familiar with finding the roots of an equation. The roots are the value(s) of $x$ when $y = 0$ .

If you graph the equation above on an $x$ - $y$ coordinate plane, you get a parabola. The parabola’s $x$ -intercepts are the roots of the equation.

Parabolas can look like either of these two graphs:

If $a$ is positive, the parabola opens upward.
If $a$ is negative, the parabola opens downward.

The vertex of each parabola is marked by the point.

diagram example of parabolas

Just like with a line, you can plug in a value for $x$ and solve for $y$ (or plug in a value for $y$ and solve for $x$ ).

Also, just like a line, the constant term at the end of the equation gives the $y$ -intercept:

$y = m x + b$
$b$ is the $y$ -intercept

$y = a x^{2} + b x + c$
$c$ is the $y$ -intercept

If curved graphs bring back memories of calculus, don’t worry - you won’t need calculus for the GRE. Most parabola questions (whether you’re given a graph or just an equation) focus on:

finding an intercept
finding a specific point on the parabola
finding where a line and a parabola intersect
shifting a parabola on the coordinate plane

You can handle all of these using algebra, much like you would with a straight line. Let’s review each one.

Finding a Point

To find a point on a parabola, substitute a number for either $x$ or $y$ , then solve for the other variable. The value you solve for, together with the value you substituted, forms an ordered pair.

If you get two solutions, that means there are two points on the parabola with the same $y$ -coordinate.

What is the value of ‘ $a$ ’ in the ordered pair $(3, a)$ that exists on the parabola $y = x^{2} + 3 x - 5 ?$

(spoiler)

Answer: $a = 13$

Finding an Intercept

To find the $x$ -intercept(s), set $y = 0$ and solve for $x$ .
To find the $y$ -intercept, set $x = 0$ and solve for $y$ .

What is the $y$ -intercept of $y = 2 x^{2} - 4 x + 3 ?$

(spoiler)

Answer: the $y$ -intercept is $3$

Intersecting Lines

To find where a parabola and a line intersect:

Set both expressions equal to $y$ .
Set the two equations equal to each other.
Solve for the $x$ -value(s) where they intersect.
Plug each $x$ -value into either equation to find the corresponding $y$ -value.

At what two points do $y = 6 x + 9$ and $y = x^{2} + 3 x + 5$ intersect?

(spoiler)

Answer: $(- 1, 3), (4, 33)$

Shifting Parabolas

Changing parts of a parabola’s equation can change its location or shape. The diagrams below show common transformations and what they do to the graph.

This isn’t commonly tested on the GRE, but it can appear.

Shifting Up or Down

diagram of parabola shifting up or down

Shifting Left or Right

diagram of parabola shifting left or right

Stretch or Flatten

diagram of parabola stretch or flatten

Flip

diagram of parabola flip

Parabolas and Their Equations

Standard form: $y = a x^{2} + b x + c$
Parabola opens upward if $a > 0$ , downward if $a < 0$
$x$ -intercepts are roots (where $y = 0$ ); $y$ -intercept is $c$ (where $x = 0$ )

Finding a Point

Substitute a value for $x$ or $y$ and solve for the other variable
Two solutions for $x$ may give two points with same $y$ -coordinate

Finding Intercepts

$x$ -intercept(s): set $y = 0$ , solve for $x$
$y$ -intercept: set $x = 0$ , solve for $y$ (equals $c$ in standard form)

Intersecting Lines and Parabolas

Set the line and parabola equations equal to each other
Solve for $x$ to find intersection points
Plug $x$ back in to find corresponding $y$ values

Shifting Parabolas

Up/Down: add/subtract constant to $y$ (affects $c$ )
Left/Right: replace $x$ with $(x \pm h)$ (shifts horizontally)
Stretch/Flatten: change value of $a$ (larger $∣ a ∣$ = steeper, smaller $∣ a ∣$ = flatter)
Flip: change sign of $a$ (positive = opens up, negative = opens down)

Graphing parabolas

You’ve probably seen many problems that include equations like this:

$y = a x^{2} + b x + c$

You may also be familiar with finding the roots of an equation. The roots are the value(s) of $x$ when $y = 0$ .

If you graph the equation above on an $x$ - $y$ coordinate plane, you get a parabola. The parabola’s $x$ -intercepts are the roots of the equation.

Parabolas can look like either of these two graphs:

If $a$ is positive, the parabola opens upward.
If $a$ is negative, the parabola opens downward.

The vertex of each parabola is marked by the point.

diagram example of parabolas

Just like with a line, you can plug in a value for $x$ and solve for $y$ (or plug in a value for $y$ and solve for $x$ ).

Also, just like a line, the constant term at the end of the equation gives the $y$ -intercept:

$y = m x + b$
$b$ is the $y$ -intercept

$y = a x^{2} + b x + c$
$c$ is the $y$ -intercept

If curved graphs bring back memories of calculus, don’t worry - you won’t need calculus for the GRE. Most parabola questions (whether you’re given a graph or just an equation) focus on:

finding an intercept
finding a specific point on the parabola
finding where a line and a parabola intersect
shifting a parabola on the coordinate plane

You can handle all of these using algebra, much like you would with a straight line. Let’s review each one.

Finding a Point

To find a point on a parabola, substitute a number for either $x$ or $y$ , then solve for the other variable. The value you solve for, together with the value you substituted, forms an ordered pair.

If you get two solutions, that means there are two points on the parabola with the same $y$ -coordinate.

What is the value of ‘ $a$ ’ in the ordered pair $(3, a)$ that exists on the parabola $y = x^{2} + 3 x - 5 ?$

(spoiler)

Answer: $a = 13$

Finding an Intercept

To find the $x$ -intercept(s), set $y = 0$ and solve for $x$ .
To find the $y$ -intercept, set $x = 0$ and solve for $y$ .

What is the $y$ -intercept of $y = 2 x^{2} - 4 x + 3 ?$

(spoiler)

Answer: the $y$ -intercept is $3$

Intersecting Lines

To find where a parabola and a line intersect:

Set both expressions equal to $y$ .
Set the two equations equal to each other.
Solve for the $x$ -value(s) where they intersect.
Plug each $x$ -value into either equation to find the corresponding $y$ -value.

At what two points do $y = 6 x + 9$ and $y = x^{2} + 3 x + 5$ intersect?

(spoiler)

Answer: $(- 1, 3), (4, 33)$

Shifting Parabolas

Changing parts of a parabola’s equation can change its location or shape. The diagrams below show common transformations and what they do to the graph.

This isn’t commonly tested on the GRE, but it can appear.