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## Digital SAT Math

# Quadratic graphs | Lesson

A guide to quadratic graphs on the digital SAT

## What are quadratic functions?

In a

**quadratic function**, the of the function is based on an expression in which the is the highest power term. For example, f, left parenthesis, x, right parenthesis, equals, x, squared, plus, 2, x, plus, 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:

- 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-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 parenthesis, 2, comma, 0, right parenthesis, then the parabola has the same y-values at x, equals, 1 and x, equals, 3, at x, equals, 0 and x, equals, 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 x, y-plane.
- Sketch a parabola that passes through the points.

**Example:**Graph f, left parenthesis, x, right parenthesis, equals, x, squared, minus, 3 in the x, y-plane.

### 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, squared-term, start color #7854ab, a, end color #7854ab, tells us whether the parabola opens upward or downward:

- If start color #7854ab, a, end color #7854ab, is greater than, 0, then the parabola opens upward.
- If start color #7854ab, a, end color #7854ab, is less than, 0, then the parabola opens downward.

The magnitude of start color #7854ab, a, end color #7854ab also describes how steep or shallow the parabola is. Parabolas with larger magnitudes of start color #7854ab, a, end color #7854ab are more steep and narrow compared to parabolas with smaller magnitudes of start color #7854ab, a, end color #7854ab, which tend to be more shallow and wide.

The graph below shows the graphs of y, equals, start color #7854ab, a, end color #7854ab, x, squared for various values of start color #7854ab, a, end color #7854ab.

The

**standard form**of a quadratic equation, y, equals, start color #7854ab, a, end color #7854ab, x, squared, plus, start color #ca337c, b, end color #ca337c, x, plus, start color #208170, c, end color #208170, shows the y-intercept of the parabola:- The y-intercept of the parabola is located at left parenthesis, 0, comma, start color #208170, c, end color #208170, right parenthesis.

The

**factored form**of a quadratic equation, y, equals, start color #7854ab, a, end color #7854ab, left parenthesis, x, minus, start color #ca337c, b, end color #ca337c, right parenthesis, left parenthesis, x, minus, start color #208170, c, end color #208170, right parenthesis, shows the x-intercept(s) of the parabola:- x, equals, start color #ca337c, b, end color #ca337c and x, equals, start color #208170, c, end color #208170 are solutions to the equation start color #7854ab, a, end color #7854ab, left parenthesis, x, minus, start color #ca337c, b, end color #ca337c, right parenthesis, left parenthesis, x, minus, start color #208170, c, end color #208170, right parenthesis, equals, 0.
- The x-intercepts of the parabola are located at left parenthesis, start color #ca337c, b, end color #ca337c, comma, 0, right parenthesis and left parenthesis, start color #208170, c, end color #208170, comma, 0, right parenthesis.
- The terms
**x-intercept**,**zero**, and**root**can be used interchangeably.

The

**vertex form**of a quadratic equation, y, equals, start color #7854ab, a, end color #7854ab, left parenthesis, x, minus, start color #ca337c, h, end color #ca337c, right parenthesis, squared, plus, start color #208170, k, end color #208170, reveals the vertex of the parabola.- The vertex of the parabola is located at left parenthesis, start color #ca337c, h, end color #ca337c, comma, start color #208170, k, end color #208170, right parenthesis.

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 f, left parenthesis, x, right parenthesis, equals, x, squared, plus, 7, x, plus, 12 ?

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 x, squared-term if necessary.

**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:

**y-intercept:**standard form**x-intercept(s):**factored form**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, equals, x, squared, minus, 4, x, minus, 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 x, y-plane. If the graph of y, equals, f, left parenthesis, x, right parenthesis is graphed in the x, y-plane and c is a positive constant:

- The graph of y, equals, f, left parenthesis, x, minus, c, right parenthesis is the graph of f, left parenthesis, x, right parenthesis shifted to the
**right**by c units. - The graph of y, equals, f, left parenthesis, x, plus, c, right parenthesis is the graph of f, left parenthesis, x, right parenthesis shifted to the
**left**by c units. - The graph of y, equals, f, left parenthesis, x, right parenthesis, plus, c is the graph of f, left parenthesis, x, right parenthesis shifted
**up**by c units. - The graph of y, equals, f, left parenthesis, x, right parenthesis, minus, c is the graph of f, left parenthesis, x, right parenthesis shifted
**down**by c units.

The graph below shows the graph of the quadratic function f, left parenthesis, x, right parenthesis, equals, x, squared, minus, 3 alongside various translations:

- The graph of start color #7854ab, f, left parenthesis, x, minus, 4, right parenthesis, equals, left parenthesis, x, minus, 4, right parenthesis, squared, minus, 3, end color #7854ab translates the graph of 4 units to the right.
- The graph of start color #ca337c, f, left parenthesis, x, plus, 6, right parenthesis, equals, left parenthesis, x, plus, 6, right parenthesis, squared, minus, 3, end color #ca337c translates the graph 6 units to the left.
- The graph of start color #208170, f, left parenthesis, x, right parenthesis, plus, 5, equals, x, squared, plus, 2, end color #208170 translates the graph 5 units up.
- The graph of start color #a75a05, f, left parenthesis, x, right parenthesis, minus, 3, equals, x, squared, minus, 6, end color #a75a05 translates the graph 3 units down.

We can also represent stretching and reflecting graphs algebraically. If the graph of y, equals, f, left parenthesis, x, right parenthesis is graphed in the x, y-plane and c is a positive constant:

- The graph of y, equals, minus, f, left parenthesis, x, right parenthesis is the graph of f, left parenthesis, x, right parenthesis reflected across the
**x-axis**. - The graph of y, equals, f, left parenthesis, minus, x, right parenthesis is the graph of f, left parenthesis, x, right parenthesis reflected across the
**y-axis**. - The graph of y, equals, c, dot, f, left parenthesis, x, right parenthesis is the graph of f, left parenthesis, x, right parenthesis stretched
**vertically**by a factor of c.

The graph below shows the graph of the quadratic function f, left parenthesis, x, right parenthesis, equals, x, squared, minus, 2, x, minus, 2 alongside various transformations:

- The graph of start color #7854ab, minus, f, left parenthesis, x, right parenthesis, equals, minus, x, squared, plus, 2, x, plus, 2, end color #7854ab is the graph of f, left parenthesis, x, right parenthesis reflected across the x-axis.
- The graph of start color #ca337c, f, left parenthesis, minus, x, right parenthesis, equals, x, squared, plus, 2, x, minus, 2, end color #ca337c is the graph of f, left parenthesis, x, right parenthesis reflected across the y-axis.
- The graph of start color #208170, 3, dot, f, left parenthesis, x, right parenthesis, equals, 3, x, squared, minus, 6, x, minus, 6, end color #208170 is the graph of f, left parenthesis, x, right parenthesis stretched vertically by a factor of 3.

### Try it!

## Your turn!

## Things to remember

### Forms of quadratic equations

**Standard form:**A parabola with the equation y, equals, start color #7854ab, a, end color #7854ab, x, squared, plus, start color #ca337c, b, end color #ca337c, x, plus, start color #208170, c, end color #208170 has its y-intercept located at left parenthesis, 0, comma, start color #208170, c, end color #208170, right parenthesis.

**Factored form:**A parabola with the equation y, equals, start color #7854ab, a, end color #7854ab, left parenthesis, x, minus, start color #ca337c, b, end color #ca337c, right parenthesis, left parenthesis, x, minus, start color #208170, c, end color #208170, right parenthesis has its x-intercept(s) located at left parenthesis, start color #ca337c, b, end color #ca337c, comma, 0, right parenthesis and left parenthesis, start color #208170, c, end color #208170, comma, 0, right parenthesis.

**Vertex form:**A parabola with the equation y, equals, start color #7854ab, a, end color #7854ab, left parenthesis, x, minus, start color #ca337c, h, end color #ca337c, right parenthesis, squared, plus, start color #208170, k, end color #208170 has its vertex located at left parenthesis, start color #ca337c, h, end color #ca337c, comma, start color #208170, k, end color #208170, right parenthesis.

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, equals, f, left parenthesis, x, right parenthesis is graphed in the x, y-plane and c is a positive constant:

- The graph of y, equals, f, left parenthesis, x, minus, c, right parenthesis is the graph of f, left parenthesis, x, right parenthesis shifted to the
**right**by c units. - The graph of y, equals, f, left parenthesis, x, plus, c, right parenthesis is the graph of f, left parenthesis, x, right parenthesis shifted to the
**left**by c units. - The graph of y, equals, f, left parenthesis, x, right parenthesis, plus, c is the graph of f, left parenthesis, x, right parenthesis shifted
**up**by c units. - The graph of y, equals, f, left parenthesis, x, right parenthesis, minus, c is the graph of f, left parenthesis, x, right parenthesis shifted
**down**by c units. - The graph of y, equals, minus, f, left parenthesis, x, right parenthesis is the graph of f, left parenthesis, x, right parenthesis reflected across the
**x-axis**. - The graph of y, equals, f, left parenthesis, minus, x, right parenthesis is the graph of f, left parenthesis, x, right parenthesis reflected across the
**y-axis**. - The graph of y, equals, c, dot, f, left parenthesis, x, right parenthesis is the graph of f, left parenthesis, x, right parenthesis stretched
**vertically**by a factor of c.

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