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## SAT (Fall 2023)

### Course: SAT (Fall 2023) > Unit 6

Lesson 4: Passport to Advanced Math: lessons by skill- Solving quadratic equations | Lesson
- Interpreting nonlinear expressions | Lesson
- Quadratic and exponential word problems | Lesson
- Manipulating quadratic and exponential expressions | Lesson
- Radicals and rational exponents | Lesson
- Radical and rational equations | Lesson
- Operations with rational expressions | Lesson
- Operations with polynomials | Lesson
- Polynomial factors and graphs | Lesson
- Graphing quadratic functions | Lesson
- Graphing exponential functions | Lesson
- Linear and quadratic systems | Lesson
- Structure in expressions | Lesson
- Isolating quantities | Lesson
- Function Notation | Lesson

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# Graphing quadratic functions | Lesson

## What are quadratic functions, and how frequently do they appear on the test?

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**.On your official SAT, you'll likely see

**2 to 4 questions**that test your understanding of the connection between quadratic functions and parabolas.**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 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

### 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?

- Is it possible to find the vertex of the parabola using the equation -b/2a as well as the other equations listed in the article?(11 votes)
- yes, it is possible, you will need to use -b/2a for the x coordinate of the vertex and another formula k=c- b^2/4a for the y coordinate of the vertex. good luck, hope this helped(7 votes)

- How do you get the formula from looking at the parabola? I am having trouble when I try to work backward with what he said.(3 votes)
- You can get the formula from looking at the graph of a parabola in two ways: Either by considering the roots of the parabola or the vertex. A parabola is not like a straight line that you can find the equation of if you have two points on the graph, because there are multiple different parabolas that can go through a given set of two points. Instead you need three points, or the vertex and a point.

In the last practice problem on this article, you're asked to find the equation of a parabola. Think about how you can find the roots of a quadratic equation by factoring. The same principle applies here, just in reverse. You can figure out the roots (x-intercepts) from the graph, and just put them together as factors to make an equation. In this form, the equation for a parabola would look like y = a(x - m)(x - n). "a" is a coefficient (responsible for vertically stretching/flipping the parabola and thus doesn't affect the roots), and the roots of the graph are at x = m and x = n. Because the graph in the problem has roots at 3 and -1, our equation would look like y = a(x + 1)(x - 3). The only one that fits this is answer choice B), which has "a" be -1.

You can also find the equation of a quadratic equation by finding the coordinates of the vertex from a graph, then plugging that into vertex form, and then picking a point on the parabola to use in order to solve for your "a" value. Thirdly, I guess you could also use three separate points to put in a system of three equations, which would let you solve for the "a", "b", and "c" in the standard form of a quadratic, but that's too much work for the SAT.(6 votes)

- my sat is on 13 of march(probably after5 days ) n i'm craming over maths I just need 500 to 600 score for math so which topics should I focus on more ??(2 votes)
- Also, remember not to stress out over it. Make sure to get a full nights. Good luck on your exam!(5 votes)

- Please help me get access to questions from "Graphing Quadratic Functions"(4 votes)
- how would i graph this though f(x)=2(x-3)^2-2(2 votes)
- Following the steps in the article, you would graph this function by following the steps to transform the parent function of y = x^2.

Here, we see that 3 is subtracted from x inside the parentheses, which means that we translate right by 3. We subtract 2 from the final answer, so we move down by 2. Our vertex will then be right 3 and down 2 from the normal vertex (0,0), at (3, -2).

From here, we see that there's a coefficient outside the parentheses, which means we vertically stretch the function by a factor of 2.

The easiest way to graph this would be to find the vertex and direction that it opens, and then plug in a point for x and see what you get for y. If we plugged in 5, we would get y = 4. You can put that point in the graph as well, and then draw a parabola that has that vertex and goes through the second point.(2 votes)