Math / Algebra

Question6. A quadratic function of the form $y=x^{2}+b x+c$ is shown graphed. (a) What are the $x$ -intercepts of this parabola? (b) Based on your answer to part (a), write the equation of this quadratic function first in factored form and then in trinomial form.

Studdy Solution

STEP 1

What is this asking? We're looking for the equation of a parabola, given its graph showing x-intercepts at $-3$ and $1$ . Watch out! Don't mix up the signs of the x-intercepts when writing the factored form.

STEP 2

1. Find the x-intercepts.
2. Write the factored form.
3. Expand to trinomial form.

STEP 3

The graph shows the parabola intersecting the x-axis at two points.
Let's find them!

STEP 4

The parabola crosses the x-axis at $x = -3$ and $x = 1$ .
These are our x-intercepts, also known as the roots or zeros of the quadratic function.
So, when $y=0$ , $x$ can be $-3$ or $1$ .

STEP 5

Knowing the x-intercepts helps us write the factored form of a quadratic equation: $y = a(x - r_1)(x - r_2)$ , where $r_1$ and $r_2$ are the roots, and $a$ determines how "stretched" or "compressed" the parabola is.

STEP 6

Plugging in our roots, we get $y = a(x - (-3))(x - 1)$ , which simplifies to $y = a(x + 3)(x - 1)$ .

STEP 7

Since the problem states the quadratic is of the form $y = x^2 + bx + c$ , we know that the coefficient of $x^2$ is $1$ .
This tells us that our stretching factor, $a$ , is also 1!
So, our factored form is $y = (x + 3)(x - 1)$ .

STEP 8

To get the trinomial form, we need to expand the factored form using the distributive property (also known as FOIL).

STEP 9

Let's multiply! $(x + 3)(x - 1) = x \cdot x + x \cdot (-1) + 3 \cdot x + 3 \cdot (-1)$ .

STEP 10

Simplifying gives us $x^2 - x + 3x - 3$ .
Combining like terms, we get $x^2 + 2x - 3$ .

STEP 11

(a) The x-intercepts are $-3$ and $1$ . (b) The factored form is $y = (x + 3)(x - 1)$ , and the trinomial form is $y = x^2 + 2x - 3$ .

Was this helpful?

Question6. A quadratic function of the form $y=x^{2}+b x+c$ is shown graphed. (a) What are the $x$ -intercepts of this parabola? (b) Based on your answer to part (a), write the equation of this quadratic function first in factored form and then in trinomial form.

Studdy Solution

STEP 1

What is this asking? We're looking for the equation of a parabola, given its graph showing x-intercepts at $-3$ and $1$ . Watch out! Don't mix up the signs of the x-intercepts when writing the factored form.

STEP 2

1. Find the x-intercepts.
2. Write the factored form.
3. Expand to trinomial form.

STEP 3

The graph shows the parabola intersecting the x-axis at two points.
Let's find them!

STEP 4

The parabola crosses the x-axis at $x = -3$ and $x = 1$ .
These are our x-intercepts, also known as the roots or zeros of the quadratic function.
So, when $y=0$ , $x$ can be $-3$ or $1$ .

STEP 5

Knowing the x-intercepts helps us write the factored form of a quadratic equation: $y = a(x - r_1)(x - r_2)$ , where $r_1$ and $r_2$ are the roots, and $a$ determines how "stretched" or "compressed" the parabola is.

STEP 6

Plugging in our roots, we get $y = a(x - (-3))(x - 1)$ , which simplifies to $y = a(x + 3)(x - 1)$ .

STEP 7

Since the problem states the quadratic is of the form $y = x^2 + bx + c$ , we know that the coefficient of $x^2$ is $1$ .
This tells us that our stretching factor, $a$ , is also 1!
So, our factored form is $y = (x + 3)(x - 1)$ .

STEP 8

To get the trinomial form, we need to expand the factored form using the distributive property (also known as FOIL).

STEP 9

Let's multiply! $(x + 3)(x - 1) = x \cdot x + x \cdot (-1) + 3 \cdot x + 3 \cdot (-1)$ .

STEP 10

Simplifying gives us $x^2 - x + 3x - 3$ .
Combining like terms, we get $x^2 + 2x - 3$ .

STEP 11

(a) The x-intercepts are $-3$ and $1$ . (b) The factored form is $y = (x + 3)(x - 1)$ , and the trinomial form is $y = x^2 + 2x - 3$ .

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question6. A quadratic function of the form y=x2+bx+cy=x^{2}+b x+cy=x2+bx+c is shown graphed. (a) What are the xxx-intercepts of this parabola? (b) Based on your answer to part (a), write the equation of this quadratic function first in factored form and then in trinomial form.

Studdy Solution

STEP 1

What is this asking? We're looking for the equation of a parabola, given its graph showing *x*-intercepts at −3-3−3 and 111. Watch out! Don't mix up the signs of the *x*-intercepts when writing the factored form.

STEP 2

1. Find the *x*-intercepts.2. Write the factored form.3. Expand to trinomial form.

STEP 3

The graph shows the parabola intersecting the *x*-axis at two points.Let's find them!

STEP 4

The parabola crosses the *x*-axis at x=−3x = -3x=−3 and x=1x = 1x=1.These are our **x-intercepts**, also known as the **roots** or **zeros** of the quadratic function.So, when y=0y=0y=0, xxx can be −3-3−3 or 111.

STEP 5

Knowing the *x*-intercepts helps us write the factored form of a quadratic equation: y=a(x−r1)(x−r2)y = a(x - r_1)(x - r_2)y=a(x−r1​)(x−r2​), where r1r_1r1​ and r2r_2r2​ are the roots, and aaa determines how "stretched" or "compressed" the parabola is.

STEP 6

Plugging in our **roots**, we get y=a(x−(−3))(x−1)y = a(x - (-3))(x - 1)y=a(x−(−3))(x−1), which simplifies to y=a(x+3)(x−1)y = a(x + 3)(x - 1)y=a(x+3)(x−1).

STEP 7

Since the problem states the quadratic is of the form y=x2+bx+cy = x^2 + bx + cy=x2+bx+c, we know that the coefficient of x2x^2x2 is 111.This tells us that our **stretching factor**, aaa, is also **1**!So, our factored form is y=(x+3)(x−1)y = (x + 3)(x - 1)y=(x+3)(x−1).

STEP 8

To get the trinomial form, we need to **expand** the factored form using the distributive property (also known as FOIL).

STEP 9

Let's multiply! (x+3)(x−1)=x⋅x+x⋅(−1)+3⋅x+3⋅(−1)(x + 3)(x - 1) = x \cdot x + x \cdot (-1) + 3 \cdot x + 3 \cdot (-1)(x+3)(x−1)=x⋅x+x⋅(−1)+3⋅x+3⋅(−1).

STEP 10

Simplifying gives us x2−x+3x−3x^2 - x + 3x - 3x2−x+3x−3.Combining like terms, we get x2+2x−3x^2 + 2x - 3x2+2x−3.

STEP 11

(a) The *x*-intercepts are −3-3−3 and 111. (b) The factored form is y=(x+3)(x−1)y = (x + 3)(x - 1)y=(x+3)(x−1), and the trinomial form is y=x2+2x−3y = x^2 + 2x - 3y=x2+2x−3.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question6. A quadratic function of the form $y=x^{2}+b x+c$ is shown graphed. (a) What are the $x$ -intercepts of this parabola? (b) Based on your answer to part (a), write the equation of this quadratic function first in factored form and then in trinomial form.

What is this asking? We're looking for the equation of a parabola, given its graph showing x-intercepts at $-3$ and $1$ . Watch out! Don't mix up the signs of the x-intercepts when writing the factored form.

1. Find the x-intercepts.
2. Write the factored form.
3. Expand to trinomial form.

The graph shows the parabola intersecting the x-axis at two points.
Let's find them!

The parabola crosses the x-axis at $x = -3$ and $x = 1$ .
These are our x-intercepts, also known as the roots or zeros of the quadratic function.
So, when $y=0$ , $x$ can be $-3$ or $1$ .

Knowing the x-intercepts helps us write the factored form of a quadratic equation: $y = a(x - r_1)(x - r_2)$ , where $r_1$ and $r_2$ are the roots, and $a$ determines how "stretched" or "compressed" the parabola is.

Plugging in our roots, we get $y = a(x - (-3))(x - 1)$ , which simplifies to $y = a(x + 3)(x - 1)$ .

Since the problem states the quadratic is of the form $y = x^2 + bx + c$ , we know that the coefficient of $x^2$ is $1$ .
This tells us that our stretching factor, $a$ , is also 1!
So, our factored form is $y = (x + 3)(x - 1)$ .

To get the trinomial form, we need to expand the factored form using the distributive property (also known as FOIL).

Let's multiply! $(x + 3)(x - 1) = x \cdot x + x \cdot (-1) + 3 \cdot x + 3 \cdot (-1)$ .

Simplifying gives us $x^2 - x + 3x - 3$ .
Combining like terms, we get $x^2 + 2x - 3$ .

(a) The x-intercepts are $-3$ and $1$ . (b) The factored form is $y = (x + 3)(x - 1)$ , and the trinomial form is $y = x^2 + 2x - 3$ .