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### Course: Digital SAT Math > Unit 12

Lesson 11: Quadratic graphs: advanced# Quadratic graphs | Lesson

A guide to quadratic graphs on the digital SAT

## What are quadratic functions?

In a $f(x)={x}^{2}+2x+1$ is a quadratic function, because in the highest power term, the $x$ is raised to the second power.

**quadratic function**, the of the function is based on an expression in which the is the highest power term. For example,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:

- Graph quadratic functions
- Identify the features of quadratic functions
- Rewrite quadratic functions to showcase specific features of graphs
- Transform quadratic functions

**You can learn anything. Let's do this!**

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

### Parabolas intro

### 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**$y$ -valueof 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 "

Parabolas also have

**vertical symmetry**along a vertical line that passes through the vertex.For example, if a parabola has a vertex at $(2,0)$ , then the parabola has the same $y$ -values at $x=1$ and $x=3$ , at $x=0$ and $x=4$ , and so on.

To graph a quadratic function:

- Evaluate the function at several different values of
.$x$ - Plot the input-output pairs as points in the
-plane.$xy$ - Sketch a parabola that passes through the points.

**Example:**Graph

### Try it!

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

### Forms & features of quadratic functions

### 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
, then the parabola opens upward.${a}>0$ - If
, then the parabola opens downward.${a}<0$

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 $y={a}{x}^{2}+{b}x+{c}$ , shows the $y$ -intercept of the parabola:

**standard form**of a quadratic equation,- The
-intercept of the parabola is located at$y$ .$(0,{c})$

The $y={a}(x-{b})(x-{c})$ ,
shows the $x$ -intercept(s) of the parabola:

**factored form**of a quadratic equation, and$x={b}$ are solutions to the equation$x={c}$ .${a}(x-{b})(x-{c})=0$ - The
-intercepts of the parabola are located at$x$ and$({b},0)$ .$({c},0)$ - The terms
, -intercept$x$ **zero**, and**root**can be used interchangeably.

The $y={a}(x-{h}{)}^{2}+{k}$ , reveals the vertex of the parabola.

**vertex form**of a quadratic equation,- The vertex of the parabola is located at
.$({h},{k})$

To identify the features of a parabola from a quadratic equation:

- Remember which equation form displays the relevant features as constants or coefficients.
- Rewrite the equation in a more helpful form if necessary.
- Identify the constants or coefficients that correspond to the features of interest.

**Example:**What are the zeros of the graph of

To match a parabola with its quadratic equation:

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

**Example:**

What is a possible equation for the parabola shown above?

### Try it!

## 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:

standard form -intercept:$y$ factored form -intercept(s):$x$ **Vertex:**vertex form

When rewriting a quadratic function to display specific graphical features:

- Choose the appropriate quadratic form based on the graphic feature to be displayed.
- 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!

## How do I transform graphs of quadratic functions?

### Intro to parabola transformations

### 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 $y=f(x)$ is graphed in the $xy$ -plane and $c$ is a positive constant:

- The graph of
is the graph of$y=f(x-c)$ shifted to the$f(x)$ **right**by units.$c$ - The graph of
is the graph of$y=f(x+c)$ shifted to the$f(x)$ **left**by units.$c$ - The graph of
is the graph of$y=f(x)+c$ shifted$f(x)$ **up**by units.$c$ - The graph of
is the graph of$y=f(x)-c$ shifted$f(x)$ **down**by units.$c$

The graph below shows the graph of the quadratic function $f(x)={x}^{2}-3$ alongside various translations:

- The graph of
translates the graph of${f(x-4)=(x-4{)}^{2}-3}$ units to the right.$4$ - The graph of
translates the graph${f(x+6)=(x+6{)}^{2}-3}$ units to the left.$6$ - The graph of
translates the graph${f(x)+5={x}^{2}+2}$ units up.$5$ - The graph of
translates the graph${f(x)-3={x}^{2}-6}$ units down.$3$

We can also represent stretching and reflecting graphs algebraically. If the graph of $y=f(x)$ is graphed in the $xy$ -plane and $c$ is a positive constant:

- The graph of
is the graph of$y=-f(x)$ reflected across the$f(x)$ . -axis$x$ - The graph of
is the graph of$y=f(-x)$ reflected across the$f(x)$ . -axis$y$ - The graph of
is the graph of$y=c\cdot f(x)$ stretched$f(x)$ **vertically**by a factor of .$c$

The graph below shows the graph of the quadratic function $f(x)={x}^{2}-2x-2$ alongside various transformations:

- The graph of
is the graph of${-f(x)=-{x}^{2}+2x+2}$ reflected across the$f(x)$ -axis.$x$ - The graph of
is the graph of${f(-x)={x}^{2}+2x-2}$ reflected across the$f(x)$ -axis.$y$ - The graph of
is the graph of${3\cdot f(x)=3{x}^{2}-6x-6}$ stretched vertically by a factor of$f(x)$ .$3$

### Try it!

## Your turn!

## Things to remember

### Forms of quadratic equations

**Standard form:**A parabola with the equation

**Factored form:**A parabola with the equation

**Vertex form:**A parabola with the equation

When we're given a quadratic function, we can rewrite the function according to the features we want to display:

standard form -intercept:$y$ factored form -intercept(s):$x$ **Vertex:**vertex form

### Transformations

If the graph of $y=f(x)$ is graphed in the $xy$ -plane and $c$ is a positive constant:

- The graph of
is the graph of$y=f(x-c)$ shifted to the$f(x)$ **right**by units.$c$ - The graph of
is the graph of$y=f(x+c)$ shifted to the$f(x)$ **left**by units.$c$ - The graph of
is the graph of$y=f(x)+c$ shifted$f(x)$ **up**by units.$c$ - The graph of
is the graph of$y=f(x)-c$ shifted$f(x)$ **down**by units.$c$ - The graph of
is the graph of$y=-f(x)$ reflected across the$f(x)$ . -axis$x$ - The graph of
is the graph of$y=f(-x)$ reflected across the$f(x)$ . -axis$y$ - The graph of
is the graph of$y=c\cdot f(x)$ stretched$f(x)$ **vertically**by a factor of .$c$

## Want to join the conversation?

- when life gives you lemons(60 votes)
- wow! nothing made sense! yay! :D SAT is in one less than month AND I DONT UNDERSTAND PARABOLAS! T^T(61 votes)
- august 26th DSAT and i dont understand transformations(6 votes)

- 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(49 votes)- Encouraging words of wisdom!(1 vote)

- Together in it, bro(9 votes)

- 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!(27 votes)
- Now I have to admit. It is the hardest chapter in entire Math section so far.(26 votes)
- tbh, along with this there are more such lessons(1 vote)

- this lesson actually has me crying. i hate quadratic functions so much :((22 votes)
- What is this transformation oh god forgive me 🤧(20 votes)
- i didn't understand ANYTHING!!(12 votes)
- Try the quadratics unit in Khan Maths.(5 votes)

- is this course really helpful to get a good sat score(10 votes)