Understanding Quadratic Expressions, Quadratic Functions, and Parabolas

Quadratic Expression
$x^{2} + 3 x + 5$
$5 x^{2} + \frac{1}{4} x + 23$
$\frac{3}{5} x^{2} + \frac{1}{2} x + 4.3$

Quadratic Expression

x^{2} + 3 x + 5

5 x^{2} + \frac{1}{4} x + 23

\frac{3}{5} x^{2} + \frac{1}{2} x + 4.3

$x$	$y$
$0$	$- 0.6$
$1$	$0.5$
$2$	$1.4$
$4$	$2.6$
$6$	$3$
$8$	$2.6$
$10$	$1.4$
$12$	$- 0.6$

x

y

0

- 0.6

1

0.5

2

1.4

4

2.6

6

3

8

2.6

10

1.4

12

- 0.6

$x$	$y$
$0$	$30$
$2$	$20$
$4$	$14$
$6$	$12$
$8$	$14$
$10$	$20$
$12$	$30$

x

y

0

30

2

20

4

14

6

12

8

14

10

20

12

30

$x$	$y$
$0$	$0$
$1$	$0.8$
$2$	$3.2$
$3$	$7.2$
$4$	$12.8$
$5$	$20$
$6$	$28.8$

x

y

0

0

1

0.8

2

3.2

3

7.2

4

12.8

5

20

6

28.8

Consider the following expression.

1 - 2 x + 3 x^{2} - 4 x^{3} + 5 - 6 x + 7 x^{2}

Combine like terms to determine whether this is a quadratic expression.

Let's recall that a quadratic expression is an expression with the highest degree equal 2. Consider the given algebraic expression. 1-2x+3x^2-4x^3+5-6x+7x^2 We can see that there are multiple combinations of like terms. Simplify this expression by combining like terms.

The highest degree of the expression is 3. This means that this is not a quadratic expression.

Consider the following expression.

21 x - 13 x^{2} + 2 x^{4} + 7 + 5 x^{2} - 2 x^{4} + 8

Combine like terms to determine whether this is a quadratic expression.

Let's recall that a quadratic expression is an expression with the highest degree equal 2. Consider the given algebraic expression. 21x-13x^2+2x^4+7+5x^2-2x^4+8 We can see that there are multiple combinations of like terms. Simplify this expression by combining like terms.

The highest degree of the expression is 2. Therefore, this is a quadratic expression.

Consider the following graphs of functions.

Which graphs correspond to quadratic functions?

We need to identify which graphs show quadratic functions. Let's recall that the graph of a quadratic function is a parabola.

Definition of a Parabola |-A parabola is a symmetrical curve where each point is equidistant from another point called the locus located between the legs of the parabola.

Now, let's analyze all the given graphs.

We can see that II is a line and not a parabola. Also, III is a horizontal curve, but it is not symmetrical. At the same time, I and IV do meet the definition of a parabola. Therefore, I and IV are the graphs of quadratic functions.

Consider the following graphs of functions.

Which graphs correspond to quadratic functions?

We need to identify which graphs show quadratic functions. Let's recall that the graph of a quadratic function is a parabola.

Definition of a Parabola |-A parabola is a symmetrical curve where each point is equidistant from another point called the locus located between the legs of the parabola.

Now, let's analyze all the given graphs.

We can see that I is a line and not a parabola. Additionally, IV is a curve whose one leg points down and another leg points up. This means that IV is also not a parabola. Conversely, II and III do meet the definition of a parabola, which means that they are the graphs of quadratic functions.

Consider a parabola.

Is the leading coefficient of the parabola positive or negative?

What are the coordinates of the vertex?

Let's consider the given parabola.

First, we need to recall the definition of the leading coefficient.

Leading Coefficient |-In a polynomial, the coefficient in front of the term with the highest degree is called the leading coefficient.

Also, we need to remember two facts.

If a leading coefficient is positive, then the parabola points upwards.
If a leading coefficient is negative, then the parabola points downwards.

In this case, the parabola points upwards, which means that its leading coefficient is positive.

We also need to determine the coordinates of the vertex of the parabola. Let's consider the parabola and find the coordinates of the lowest point.

Therefore, the coordinates of the vertex are (1,- 4).

Consider a parabola.

Is the leading coefficient of the parabola positive or negative?

What are the coordinates of the vertex?

Let's consider the given parabola.

First, we need to recall the definition of the leading coefficient.

Leading Coefficient |-In a polynomial, the coefficient in front of the term with the highest degree is called the leading coefficient.

Also, we need to remember two facts.

If a leading coefficient is positive, then the parabola points upwards.
If a leading coefficient is negative, then the parabola points downwards.

In this case, the parabola points downwards, which means that its leading coefficient is negative.

We also need to determine the coordinates of the vertex of the parabola. Let's consider the parabola and find the coordinates of the highest point.

Therefore, the coordinates of the vertex are (2,3).

Quadratic Functions

Catch-Up and Review

Analyzing Different Functions

Introducing Quadratic Expressions

Quadratic Expression

Following a Quadratic Path to a New Beginning

Hint

Solution

Identifying a Quadratic Expression

Parabola

Direction of a Parabola

Quadratic Function

Identifying a Parabola

Determining the Sign of a Leading Coefficient

Building a Bridge

Answer

Hint

Solution

Kangaroo Population

Answer

Hint

Solution

Setting up a Communication System on Harmonica

Hint

Solution

Analyzing Horizontal Parabolas

Quadratic Functions

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