Zeros of a Fuction - Examples, Exercises and Solutions

To solve for the x-intercepts of the function $f(x) = -x^2 + 5x + 6$, we will find the roots of the quadratic equation $-x^2 + 5x + 6 = 0$.

Let's attempt to factor this quadratic equation first. Rewrite the equation as follows:

$-x^2 + 5x + 6 = 0$.

To factor, we look for two numbers that multiply to $-6$ (the product of $a$ and $c$, where $a = -1$ and $c = 6$) and add to $5$ (the middle coefficient $b$).

The numbers that satisfy this condition are $-1$ and $6$.

Thus, the quadratic can be factored as:

$(x - 6)(x + 1) = 0$.

Setting each factor equal to zero gives us:

$x - 6 = 0$ or $x + 1 = 0$.

Solving these equations, we find:

$x = 6$ and $x = -1$.

Thus, the points A and B, the x-intercepts of the function, are:

$(-1, 0)$ and $(6, 0)$.

Therefore, the solution to the problem is $(-1, 0), (6, 0)$.

To solve for the points A and B, we need to find the roots of the function $f(x) = x^2 - 6x + 5$ where $f(x) = 0$.

Let's proceed step-by-step:

• Step 1: Set the function to zero
  We begin by setting the equation to zero: $x^2 - 6x + 5 = 0$.
• Step 2: Factor the quadratic
  We need to factor the expression. We look for two numbers that multiply to $c = 5$ and add to $b = -6$. These numbers are $-1$ and $-5$.
• Step 3: Write the factorization
  Therefore, we can write the quadratic as: $(x - 1)(x - 5) = 0$.
• Step 4: Solve for the roots
  Set each factor equal to zero: \begin{align*} x - 1 &= 0 \\ x &= 1 \end{align*} \begin{align*} x - 5 &= 0 \\ x &= 5 \end{align*} The roots are $x = 1$ and $x = 5$.
• Step 5: Identify the Points A and B
  The points A and B, where the function intersects the x-axis, are $(1, 0)$ and $(5, 0)$.

Thus, the coordinates of points A and B are $(1,0),(5,0)$, which matches choice 1.

Let's solve the problem by following the outlined analysis:

• Step 1: Identify important points on the parabola.
• Step 2: Calculate the y-intercept by evaluating $f(0)$.
• Step 3: Confirm understanding of the vertex form and its characteristics.

Step 1: Identify the significant points on the function.
The function given is $f(x) = x^2 - 8x + 16$.

This function can be seen as
$f(x) = (x - 4)^2$.

This format not only indicates it is always non-negative but also reveals the vertex is located at $x = 4$, importantly, with $f(x) = 0$.

Step 2: Calculate the y-intercept.
Evaluate the function at $x = 0$:

$f(0) = 0^2 - 8 \cdot 0 + 16 = 16$.
So, the y-intercept is $(0, 16)$.

Thus, point A, which is often labeled at a crucial intercept, corresponds to the y-intercept of the function. The calculation confirms that point A is $(0, 16)$.

Therefore, the solution to the problem is $(0,16)$.

To solve the exercise, first note that point C lies on the X-axis.

Therefore, to find it, we need to understand what is the X value when Y equals 0.

Let's set the equation equal to 0:

0=x²-8x+16

We'll use the preferred method (trinomial or quadratic formula) to find the X values, and we'll discover that

X=4

To solve for points A and B, we find the x-intercepts of the function by setting:

$f(x) = x^2 - 3x - 4 = 0$

We check if it can be factored:

Factor $x^2 - 3x - 4$. The factors of -4 that add to -3 are -4 and 1.

Thus, factor the function as $(x - 4)(x + 1) = 0$.

Set each factor to zero:

• $x - 4 = 0$ implies $x = 4$
• $x + 1 = 0$ implies $x = -1$

These are the x-intercepts, or roots, of the quadratic function.

Therefore, the coordinates of points A and B, where the function intersects the x-axis, are $A(-1, 0)$ and $B(4, 0)$.

The correct choice corresponding to these points is option 3: $A(-1,0), B(4,0)$.

Thus, the solution to the problem is $A(-1,0), B(4,0)$.

To solve for the points A and B, where the graph of $f(x) = x^2 - 6x$ intersects the x-axis, let's solve the equation $f(x) = 0$:
1. Write the equation in standard form:
$x^2 - 6x = 0$.

2. Factor the quadratic equation:
$x(x - 6) = 0$.

3. Set each factor equal to zero:
$x = 0$ or $x - 6 = 0$.

4. Solve each equation for $x$:
$x = 0$ and $x = 6$.

Thus, the points where the function intersects the x-axis, also the points A and B, are $(0,0)$ and $(6,0)$.

Therefore, the solution to the problem is $(6,0), (0,0)$.

To solve the problem of finding points A and B on the graph of the function $f(x) = x^2 - 6x + 8$, we need to determine where this quadratic function equals zero.

Step-by-step Approach:

• Step 1: Check Factoring Possibility
  The quadratic function is given by:

We will attempt to factor this quadratic expression. We are looking for two numbers that multiply to the constant term, 8, and add to the coefficient of $x$, which is $-6$.

• Step 2: Identify Factors
  The numbers $4$ and $2$ multiply to $8$ and add to $-6$, if we consider their negative counterparts, $-4$ and $-2$:

This matches our expression, confirming that it is the correct factorization.

• Step 3: Solve for Roots
  Set each factor equal to zero:

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

Therefore, the solution to the problem is that points A and B are at $(2,0)$ and $(4,0)$.

Final Solution:
The points A and B are $(2,0)$ and $(4,0)$.

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 $(-1, 0), (-2, 0)$.