Product Representation: Linking function properties to its representation

To determine the points of intersection of the function $y=(x-3)(x+3)$ with the x-axis, we need to set $y$ to zero and solve for $x$.

Follow these steps:

• Step 1: Set the function equal to zero: $(x-3)(x+3) = 0$.
• Step 2: Apply the zero-product property, solving each factor for zero:
• For $x-3=0$:

$x = 3$

• For $x+3=0$:

$x = -3$

Thus, the points of intersection of the function with the x-axis, or the x-intercepts, are $(-3, 0)$ and $(3, 0)$.

Therefore, the solution to the problem, confirming x-intercepts, is $(-3, 0), (3, 0)$.

In order to find the point of the intersection with the X-axis, we first need to establish that Y=0.

0 = (x-5)(x+5)

When we have an equation of this type, we know that one of these parentheses must be equal to 0, so we begin by checking the possible options.

x-5 = 0
x = 5

x+5 = 0
x = -5

That is, we have two points of intersection with the x-axis, when we discover their x points, and the y point is already known to us (0, as we placed it):

(5,0)(-5,0)

This is the solution!

To determine the points of intersection with the x-axis for the function $y = x(x+5)$, follow these steps:

• Step 1: Set the function equal to zero to find the x-intercepts: $y = 0$.
• Step 2: Solve the equation $x(x+5) = 0$.

Considering the product $x(x+5) = 0$:

• If $x = 0$, then one solution is $x = 0$.
• If $x+5 = 0$, then solving for $x$ gives $x = -5$.

Thus, the two points of intersection with the x-axis are:

$(-5, 0)$ and $(0, 0)$.

Therefore, the points of intersection of the function $y = x(x+5)$ with the x-axis are $(-5, 0)$ and $(0, 0)$.

To find the points of intersection, follow these steps:

• Step 1: The function given is $y = (x+7)(x+2)$. We are interested in where this function intersects the x-axis, which occurs when $y = 0$.
• Step 2: Set the function equal to zero: $(x+7)(x+2) = 0$.
• Step 3: Solve for $x$ using the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.

Now, solve the equation:
Step 1: Set $x+7 = 0$, which gives $x = -7$.
Step 2: Set $x+2 = 0$, which gives $x = -2$.

These values are the $x$-coordinates where the function intersects the x-axis. Since the y-coordinates at each of these points is zero, the intersection points are $(-7,0)$ and $(-2,0)$.

Therefore, the points of intersection are $(-2,0)$ and $(-7,0)$.

To determine the points of intersection of the function $y = (x+3)(x-3)$ with the x-axis, we need to find the x-values where $y = 0$. These are called the x-intercepts.

We begin by setting the function equal to zero:

$(x+3)(x-3) = 0$

Using the zero-product property, if a product of two terms is zero, then at least one of the factors must be zero. Thus, we set each factor equal to zero and solve for $x$:

• First factor: $x + 3 = 0$
• Solving for $x$, we subtract 3 from both sides:
  $x = -3$
• Second factor: $x - 3 = 0$
• Solving for $x$, we add 3 to both sides:
  $x = 3$

Hence, the solutions for $x$ where $y = 0$ are $x = -3$ and $x = 3$.

Therefore, the points of intersection of the function with the x-axis are $(-3, 0)$ and $(3, 0)$.

Comparing with the given answer choices, the correct choice is $(3,0),(-3,0)$.

Therefore, the points of intersection are $(3,0),(-3,0)$.

To determine the points where the function intersects the x-axis, we need to find the x-intercepts. These occur where $y = 0$.

The function is given as $y = (x-11)(x+1)$. To find the x-intercepts, we set this function equal to zero:

$(x-11)(x+1) = 0$.

This equation implies that the product is zero when either $x-11 = 0$ or $x+1 = 0$.

Solving these equations, we find:

• $x-11 = 0$ leads to $x = 11$.
• $x+1 = 0$ leads to $x = -1$.

Thus, the points of intersection with the x-axis are $(-1, 0)$ and $(11, 0)$.

Therefore, the solution to the problem is $(-1, 0), (11, 0)$.

The solution to the problem involves finding the x-intercepts of the given quadratic function, which are the points where the function intersects the x-axis (i.e., where $y = 0$).

Step by step solution:

• Set the function equal to zero: $(x+8)(x-9) = 0$.
• The product of two terms is zero if and only if at least one of the terms is zero. Therefore, we solve:
  $x+8 = 0$ or $x-9 = 0$.
• For $x+8 = 0$:
  Subtract 8 from both sides to isolate $x$:
  $x = -8$.
• For $x-9 = 0$:
  Add 9 to both sides to isolate $x$:
  $x = 9$.

The function intersects the x-axis at the points $(-8, 0)$ and $(9, 0)$.

Therefore, the points of intersection of the function with the x-axis are $(-8, 0)$ and $(9, 0)$.

To solve this problem, we'll follow these steps:

• Step 1: Set the function $y = (4x + 8)(x + 1)$ equal to zero, i.e., $(4x + 8)(x + 1) = 0$.
• Step 2: Apply the zero-product property to resulting factors.
• Step 3: Solve each equation for $x$.

Now, let's work through each step:
Step 1: We start by setting the equation to zero:
$(4x + 8)(x + 1) = 0$.

Step 2: Using the zero-product property, we find:
1. $4x + 8 = 0$
2. $x + 1 = 0$

Step 3: Solve each of these equations for $x$:

For $4x + 8 = 0$, subtract 8 from both sides to get $4x = -8$. Divide both sides by 4, resulting in:
$x = -2$.

For $x + 1 = 0$, subtract 1 from both sides to get:
$x = -1$.

Thus, the points of intersection of the function with the x-axis are the solutions we just found. At these points, the y-value is zero, giving us the intersection points as $(-1, 0)$ and $(-2, 0)$.

Therefore, the solution to the problem is $(-1, 0), (-2, 0)$.

To solve this problem, we'll determine the intersection points of the function $y = (x - 1)(x - 1)$ with the x-axis by following these steps:

• Step 1: Set the function equal to zero to find where it intersects the x-axis.
• Step 2: Solve the equation $(x - 1)^2 = 0$.
• Step 3: Determine the x-coordinate(s) from this equation.
• Step 4: The y-coordinate will be zero at these points.

Let's work through these steps:

Step 1: We set the given function to zero: $y = (x - 1)^2 = 0$.

Step 2: By solving the equation $(x - 1)^2 = 0$, we apply the property that a square is zero only if the base is zero.

Step 3: Solving $(x - 1) = 0$, we find:

$x = 1$

Step 4: The corresponding point on the graph is $(1, 0)$, indicating where the function crosses the x-axis.

Therefore, the point of intersection of the function with the x-axis is $(1, 0)$.

To solve for the x-intercepts of the function $y = x(-x-1)$, we set $y$ to zero and solve the equation $x(-x-1) = 0$.

Step 1: Identify that the equation is already factored. Set each factor equal to zero:

• $x = 0$
• $-x-1 = 0$

Step 2: Solve for $x$ in each case:
For $x = 0$, the solution is $x = 0$.
For $-x-1 = 0$, add 1 to both sides to get $-x = 1$, then multiply by -1 to find $x = -1$.

Thus, the points of intersection with the x-axis are at $x = 0$ and $x = -1$.

Final Coordinates: Because these are x-intercepts, for both points, the y-coordinate is 0. Therefore, the points of intersection are $(-1, 0)$ and $(0, 0)$.

The correct choice from the given options is $( -1, 0 ), ( 0, 0 )$.

To solve this problem, we'll need to determine where the function intersects the x-axis, which occurs where $y = 0$.

Let's work through the solution:

• Step 1: Set the function equal to zero: $x(x-1) = 0$.
• Step 2: Apply the zero-product property, which states that if the product of two numbers is zero, at least one of the factors must be zero.
• Step 3: Set each factor equal to zero: $x = 0$ or $x - 1 = 0$.
• Step 4: Solve for $x$ in each equation:
  - For $x = 0$: This immediately gives us the solution $x = 0$.
  - For $x-1 = 0$: Add 1 to both sides to get $x = 1$.

Therefore, the two points of intersection with the x-axis are $(0,0)$ and $(1,0)$.

This matches with choice 2, thus confirming the correct option.

Thus, the points of intersection with the x-axis are $(0,0)$ and $(1,0)$.

To solve this problem, we'll follow these detailed steps:

• Step 1: Recognize that to find the intersection with the x-axis, we set $y = 0$ for the function $y = (x-2)(x+3)$.
• Step 2: Solve the equation $(x-2)(x+3) = 0$.
• Step 3: Use the zero-product property, which states that if a product equals zero, then at least one of the factors must be zero. Thus:
• $x-2 = 0$ or $x+3 = 0$
• Step 4: Solve each equation:
• For $x-2 = 0$, we add 2 to both sides, yielding $x = 2$.
• For $x+3 = 0$, we subtract 3 from both sides, yielding $x = -3$.
• Step 5: These x-values represent the points of intersection with the x-axis, or the x-intercepts.

Thus, the points of intersection of the function
$y = (x-2)(x+3)$ with the x-axis are the coordinates $(-3,0)$ and $(2,0)$.

Therefore, the solution to the problem is the points $(-3,0),(2,0)$.

To find where the function intersects the x-axis, we set $y = (x - 1)(x + 10) = 0$.

Using the Zero Product Property, if the product equals zero, at least one of the factors must be zero:

• If $(x - 1) = 0$, then $x = 1$.
• If $(x + 10) = 0$, then $x = -10$.

Thus, the function intersects the x-axis at the points where $x = 1$ and $x = -10$. These give us the points $(1, 0)$ and $(-10, 0)$ respectively, as the y-coordinate is zero for all x-intercepts.

Therefore, the points of intersection are $(1, 0)$ and $-10, 0)$.

To determine if the function $y = (x-2)(x+1)$ has a minimum or maximum point, we start by converting it from product form to standard form:

$y = (x-2)(x+1)$

Expanding the expression:

$y = x^2 + x - 2x - 2$

Simplify:

$y = x^2 - x - 2$

In standard form, $y = x^2 - x - 2$, the coefficient of $x^2$, which is $a = 1$, is positive. A positive $a$ indicates the parabola opens upwards.

Since the parabola opens upwards, it has a minimal point (vertex) as its lowest point.

Therefore, the parabola $y = (x-2)(x+1)$ has a minimal point.

To determine the points of intersection of the function $y = 2x(2x+4)$ with the x-axis, we must find where the function equals zero. Such points occur where $y = 0$.

Start by setting the equation to zero:

$2x(2x + 4) = 0$

Using the zero-product property, which states that if a product of multiple factors is zero, then at least one of the factors must be zero, we solve as follows:

• Set each factor to zero: $2x = 0$ and $2x + 4 = 0$.
• Solving $2x = 0$: Divide both sides by 2 to get $x = 0$.
• Solving $2x + 4 = 0$: Subtract 4 from both sides to get $2x = -4$, then divide by 2 to get $x = -2$.

Thus, the x-intercepts are at $x = 0$ and $x = -2$.

Correspondingly, the points of intersection are $(0, 0)$ and $(-2, 0)$.

By comparing with the provided multiple-choice options, the correct answer is indeed choice 1: $(-2,0),(0,0)$.

To solve this problem, we will find the x-intercepts of the function $y = (x-2)(x+4)$.

The function is already in factored form: $y = (x-2)(x+4)$. The x-intercepts occur where $y = 0$.

Set the equation equal to zero:

$(x-2)(x+4) = 0$

Using the Zero Product Property, each factor must equal zero:

• First solve $x - 2 = 0$:
• Add 2 to both sides: $x = 2$
• Next, solve $x + 4 = 0$:
• Subtract 4 from both sides: $x = -4$

The x-intercepts of the function are at points $(2, 0)$ and $(-4, 0)$.

Thus, the points at which the function intersects the x-axis are $(-4,0)$ and $(2,0)$.

Therefore, the correct answer is choice 3: $(-4,0),(2,0)$.

To solve this problem, we'll determine where the function intersects the x-axis by following these steps:

• Step 1: Set the function equal to zero to find the roots.
• Step 2: Solve each factor of the equation independently for $x$.
• Step 3: Confirm the intersection points as the solutions.

Now, let's work through each step:
Step 1: Given the function $y = (x-9)(x+7)$, set $y = 0$ to find the x-intercepts:
$(x-9)(x+7) = 0$.

Step 2: Solve the equation:
The expression $(x-9)(x+7) = 0$ implies that either $x-9 = 0$ or $x+7 = 0$.

Solving each equation:
For $x-9 = 0$, solve for $x$:
$x = 9$.
For $x+7 = 0$, solve for $x$:
$x = -7$.

Step 3: Therefore, the points of intersection are where $y = 0$, which occur at:

The solutions are $x = 9$ and $x = -7$.

The coordinates of these intersection points, given $y = 0$ at each root, are $(-7, 0)$ and $(9, 0)$.

Therefore, the solution to the problem is $(-7, 0), (9, 0)$.

To solve the problem of finding the intersection points of the function $y = x(x + 1)$ with the x-axis, follow these steps:

• Step 1: Understand that the function intersects the x-axis where $y = 0$.
• Step 2: Set up the equation $x(x + 1) = 0$.
• Step 3: Solve each part of the product for zero:
• For $x = 0$, the first solution is $x = 0$.
• For $x + 1 = 0$, solving gives us the second solution $x = -1$.

These solutions, $x = 0$ and $x = -1$, correspond to the points $(-1, 0)$ and $(0, 0)$ on the Cartesian plane. Thus, the points of intersection are $(-1, 0)$ and $(0, 0)$.

The correct choice from the provided options is:

• : $(-1, 0), (0, 0)$

Therefore, the solution to the problem is that the function $y = x(x + 1)$ intersects the x-axis at $( -1, 0 ), ( 0, 0 )$.

To find the points of intersection of the function $y = (x+3)(4x-4)$ with the x-axis, we set $y = 0$ and solve for $x$.

First, we set each factor of the expression to zero:

• For the first factor, $x+3 = 0$:
• Solve for $x$:
• $x + 3 = 0$
• $x = -3$
• For the second factor, $4x-4 = 0$:
• Solve for $x$:
• $4x - 4 = 0$
• $4x = 4$
• $x = 1$

The points of intersection are where these $x$ values occur with $y = 0$. Thus, the points are $(-3, 0)$ and $(1, 0)$.

Therefore, the solution to the problem is $(-3,0),(1,0)$.

To solve for the points of intersection of the function $y = (x-2)(x+4)$ with the x-axis, we proceed as follows:

• Set the function equal to zero to find the x-intercepts: $(x-2)(x+4) = 0$.
• Apply the zero-product property, which tells us that if a product of factors equals zero, then at least one of the factors must be zero. Thus, we solve the equations:
• $x-2 = 0$ or $x+4 = 0$.

Solving these equations, we find:

$x-2 = 0$ gives $x = 2$.

$x+4 = 0$ gives $x = -4$.

Therefore, the points of intersection with the x-axis are the points where $y=0$. Substituting these x-values into $y = (x-2)(x+4)$, we confirm that the corresponding y-values are zero:

• For $x = 2$, the point is $(2,0)$.
• For $x = -4$, the point is $(-4,0)$.

Thus, the points of intersection are $(-4,0)$ and $(2,0)$.