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### Course: Digital SAT Math>Unit 12

A guide to quadratic graphs on the digital SAT

of the function is based on an expression in which the
is the highest power term. For example, $f\left(x\right)={x}^{2}+2x+1$ is a quadratic function, because in the highest power term, the $x$ is raised to the second power.
Unlike the graphs of linear functions, the graphs of quadratic functions are nonlinear: they don't look like straight lines. Specifically, the graphs of quadratic functions are called parabolas.
In this lesson, we'll learn to:
2. Identify the features of quadratic functions
3. Rewrite quadratic functions to showcase specific features of graphs
You can learn anything. Let's do this!

## How do I graph parabolas, and what are their features?

### Parabolas intro

Parabolas introSee video transcript

### What are the features of a parabola?

All parabolas have a $y$-intercept, a
, and open either upward or downward.
Since the vertex is the point at which a parabola changes from increasing to decreasing or vice versa, it is also either the maximum or minimum $y$-value of the parabola.
• If the parabola opens upward, then the vertex is the lowest point on the parabola.
• If the parabola opens downward, then the vertex is the highest point on the parabola.
A parabola can also have zero, one, or two $x$-intercepts.
Note: the terms "zero" and "root" are used interchangeably with "$x$-intercept". They all mean the same thing!
Parabolas also have vertical symmetry along a vertical line that passes through the vertex.
For example, if a parabola has a vertex at $\left(2,0\right)$, then the parabola has the same $y$-values at $x=1$ and $x=3$, at $x=0$ and $x=4$, and so on.
1. Evaluate the function at several different values of $x$.
2. Plot the input-output pairs as points in the $xy$-plane.
3. Sketch a parabola that passes through the points.

Example: Graph $f\left(x\right)={x}^{2}-3$ in the $xy$-plane.

### Try it!

TRY: match the features of the parabola to their coordinates
The graph of $y=-{x}^{2}+4x+5$ is shown above. Match the features of the graph with their coordinates.

## How do I identify features of parabolas from quadratic functions?

### Forms & features of quadratic functions

Forms & features of quadratic functionsSee video transcript

### Standard form, factored form, and vertex form: What forms do quadratic equations take?

For all three forms of quadratic equations, the coefficient of the ${x}^{2}$-term, $a$, tells us whether the parabola opens upward or downward:
• If $a>0$, then the parabola opens upward.
• If $a<0$, then the parabola opens downward.
The magnitude of $a$ also describes how steep or shallow the parabola is. Parabolas with larger magnitudes of $a$ are more steep and narrow compared to parabolas with smaller magnitudes of $a$, which tend to be more shallow and wide.
The graph below shows the graphs of $y=a{x}^{2}$ for various values of $a$.
The standard form of a quadratic equation, $y=a{x}^{2}+bx+c$, shows the $y$-intercept of the parabola:
• The $y$-intercept of the parabola is located at $\left(0,c\right)$.
The factored form of a quadratic equation, $y=a\left(x-b\right)\left(x-c\right)$, shows the $x$-intercept(s) of the parabola:
• $x=b$ and $x=c$ are solutions to the equation $a\left(x-b\right)\left(x-c\right)=0$.
• The $x$-intercepts of the parabola are located at $\left(b,0\right)$ and $\left(c,0\right)$.
• The terms $x$-intercept, zero, and root can be used interchangeably.
The vertex form of a quadratic equation, $y=a\left(x-h{\right)}^{2}+k$, reveals the vertex of the parabola.
• The vertex of the parabola is located at $\left(h,k\right)$.
To identify the features of a parabola from a quadratic equation:
1. Remember which equation form displays the relevant features as constants or coefficients.
2. Rewrite the equation in a more helpful form if necessary.
3. Identify the constants or coefficients that correspond to the features of interest.

Example: What are the zeros of the graph of $f\left(x\right)={x}^{2}+7x+12$ ?

To match a parabola with its quadratic equation:
1. Determine the features of the parabola.
2. Identify the features shown in quadratic equation(s).
3. Select a quadratic equation with the same features as the parabola.
4. Plug in a point that is not a feature from Step 2 to calculate the coefficient of the ${x}^{2}$-term if necessary.

Example:
What is a possible equation for the parabola shown above?

### Try it!

TRY: determine the feature of a parabola from its equation
The graph of the equation $y=\frac{1}{2}{x}^{2}-7$ is a parabola that opens
because the coefficient of ${x}^{2}$ is
.

TRY: determine the feature of a parabola from its equation
The
of the graph of $f\left(x\right)=\left(x+2{\right)}^{2}+7$ is located at the point $\left(-2,7\right)$. $7$ is the
$y$-value of the graph.

TRY: determine the feature of a parabola from its equation
$y=\left(x-2\right)\left(x-4\right)$ is the
form of a quadratic equation. The
of the graph can be identified as constants in the equation.

## How do I rewrite quadratic functions to reveal specific features of parabolas?

### Equivalent forms of quadratic functions

When we're given a quadratic function, we can rewrite the function according to the features we want to display:
• $y$-intercept: standard form
• $x$-intercept(s): factored form
• Vertex: vertex form
When rewriting a quadratic function to display specific graphical features:
1. Choose the appropriate quadratic form based on the graphic feature to be displayed.
2. Rewrite the given quadratic expression as an equivalent expression in the form identified in Step 1.

Example:
The graph of $y={x}^{2}-4x-5$ is shown above. Write an equivalent equation from which the coordinates of the vertex can be identified as constants in the equation.

### Try it!

Try: identify the appropriate forms to use
A graph of the quadratic equation $y={x}^{2}-2x-3$ is shown in the $xy$-plane above.
The vertex of the parabola is located at
. To show the coordinates of the vertex as constants or coefficients, we should use the
form of the quadratic equation, $y=\left(x-1{\right)}^{2}-4$.
The $x$-intercepts of the parabola are
. To show the $x$-intercepts as constants or coefficients, we should use the
form of the quadratic equation, $y=\left(x+1\right)\left(x-3\right)$.

## How do I transform graphs of quadratic functions?

### Intro to parabola transformations

Intro to parabola transformationsSee video transcript

### Translating, stretching, and reflecting: How does changing the function transform the parabola?

We can use function notation to represent the translation of a graph in the $xy$-plane. If the graph of $y=f\left(x\right)$ is graphed in the $xy$-plane and $c$ is a positive constant:
• The graph of $y=f\left(x-c\right)$ is the graph of $f\left(x\right)$ shifted to the right by $c$ units.
• The graph of $y=f\left(x+c\right)$ is the graph of $f\left(x\right)$ shifted to the left by $c$ units.
• The graph of $y=f\left(x\right)+c$ is the graph of $f\left(x\right)$ shifted up by $c$ units.
• The graph of $y=f\left(x\right)-c$ is the graph of $f\left(x\right)$ shifted down by $c$ units.
The graph below shows the graph of the quadratic function $f\left(x\right)={x}^{2}-3$ alongside various translations:
• The graph of $f\left(x-4\right)=\left(x-4{\right)}^{2}-3$ translates the graph of $4$ units to the right.
• The graph of $f\left(x+6\right)=\left(x+6{\right)}^{2}-3$ translates the graph $6$ units to the left.
• The graph of $f\left(x\right)+5={x}^{2}+2$ translates the graph $5$ units up.
• The graph of $f\left(x\right)-3={x}^{2}-6$ translates the graph $3$ units down.
We can also represent stretching and reflecting graphs algebraically. If the graph of $y=f\left(x\right)$ is graphed in the $xy$-plane and $c$ is a positive constant:
• The graph of $y=-f\left(x\right)$ is the graph of $f\left(x\right)$ reflected across the $x$-axis.
• The graph of $y=f\left(-x\right)$ is the graph of $f\left(x\right)$ reflected across the $y$-axis.
• The graph of $y=c\cdot f\left(x\right)$ is the graph of $f\left(x\right)$ stretched vertically by a factor of $c$.
The graph below shows the graph of the quadratic function $f\left(x\right)={x}^{2}-2x-2$ alongside various transformations:
• The graph of $-f\left(x\right)=-{x}^{2}+2x+2$ is the graph of $f\left(x\right)$ reflected across the $x$-axis.
• The graph of $f\left(-x\right)={x}^{2}+2x-2$ is the graph of $f\left(x\right)$ reflected across the $y$-axis.
• The graph of $3\cdot f\left(x\right)=3{x}^{2}-6x-6$ is the graph of $f\left(x\right)$ stretched vertically by a factor of $3$.

### Try it!

TRY: Shift a parabola
Compared to the graph of $y={x}^{2}$, the graph of $y=\left(x+4{\right)}^{2}+3$ is shifted $4$ units
and $3$ units
.

Try: reflect a parabola
The graph of $y=f\left(x\right)$ is a parabola that opens upward and has a vertex located at $\left(2,1\right)$.
The graph of $y=f\left(-x\right)$ has a vertex located at
.
The graph of $y=-f\left(x\right)$ is a parabola that opens
.

Practice: identify the features of a parabola
If the function $f\left(x\right)=\left(x-3{\right)}^{2}-11$ is graphed in the $xy$-plane, what are the coordinates of the vertex?

Practice: match a parabola to a quadratic function
Function $f$ is graphed in the $xy$-plane above. Which of the following could be $f$ ?

Practice: identify the graphical feature displayed as a constant or coefficient
If $y=\left(x-2{\right)}^{2}+6$ is graphed in the $xy$-plane. which of the following characteristics of the graph is displayed as a constant or coefficient in the equation?

## Things to remember

Standard form: A parabola with the equation $y=a{x}^{2}+bx+c$ has its $y$-intercept located at $\left(0,c\right)$.
Factored form: A parabola with the equation $y=a\left(x-b\right)\left(x-c\right)$ has its $x$-intercept(s) located at $\left(b,0\right)$ and $\left(c,0\right)$.
Vertex form: A parabola with the equation $y=a\left(x-h{\right)}^{2}+k$ has its vertex located at $\left(h,k\right)$.
When we're given a quadratic function, we can rewrite the function according to the features we want to display:
• $y$-intercept: standard form
• $x$-intercept(s): factored form
• Vertex: vertex form

### Transformations

If the graph of $y=f\left(x\right)$ is graphed in the $xy$-plane and $c$ is a positive constant:
• The graph of $y=f\left(x-c\right)$ is the graph of $f\left(x\right)$ shifted to the right by $c$ units.
• The graph of $y=f\left(x+c\right)$ is the graph of $f\left(x\right)$ shifted to the left by $c$ units.
• The graph of $y=f\left(x\right)+c$ is the graph of $f\left(x\right)$ shifted up by $c$ units.
• The graph of $y=f\left(x\right)-c$ is the graph of $f\left(x\right)$ shifted down by $c$ units.
• The graph of $y=-f\left(x\right)$ is the graph of $f\left(x\right)$ reflected across the $x$-axis.
• The graph of $y=f\left(-x\right)$ is the graph of $f\left(x\right)$ reflected across the $y$-axis.
• The graph of $y=c\cdot f\left(x\right)$ is the graph of $f\left(x\right)$ stretched vertically by a factor of $c$.

## Want to join the conversation?

• when life gives you lemons
• eat them
• wow! nothing made sense! yay! :D SAT is in one less than month AND I DONT UNDERSTAND PARABOLAS! T^T
• august 26th DSAT and i dont understand transformations
• guys i would recommend searching desmos calculator on google and opening it on other tab, because it is used for graphing and it is also used for digital SAT hence it is very useful to get a perfect score in your math section
all the best lads yall got this
• Encouraging words of wisdom!
(1 vote)
• oct 7
• Together in it, bro
• For those, who face difficulty to understand this lesson - Please search for extra content on Youtube. This lesson needs lots of background knowledge. So, whichever term or topic you feel is tough search on Youtube and gain some basic knowledge about that. It takes me 2 days to complete this lesson! But eventually, I understand everything :) Try hard, never give up!
• Now I have to admit. It is the hardest chapter in entire Math section so far.
• tbh, along with this there are more such lessons
(1 vote)
• this lesson actually has me crying. i hate quadratic functions so much :(