Finding the Zeros of a Parabola

Finding the zeros of a quadratic function of the form $y=ax^2+bx+c$

Zero points of a function are its intersection points with the $X$ -axis.
To find them, we set $Y=0$ ,
we get an equation that can sometimes be solved using a trinomial or the quadratic formula.

When trying to find the zero point, you can encounter three possible results:

Two results -
In this case, the function intersects the $X$ -axis at two different points.
One result -
In this case, the function intersects the $X$ -axis at only one point, meaning the vertex of the parabola is exactly on the $X$ -axis.
No results -
In this case, the function does not intersect the $X$ -axis at all, meaning it hovers above or below it.

Detailed explanation

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Test yourself on zeros of a fuction!

The following function has been graphed below:

$$ f(x)=-x^2+5x+6 $$

Calculate points A and B.

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Zero points describe a situation where the function equals zero.

Let's look at an example:
We have the function
$X^2-4x+5$
We substitute into the quadratic formula and get:

${-4 \pm \sqrt{(-4)^2-4*1*5} \over 21}=-\frac{4\pm\sqrt{-4}}{2}$

This equation has no solution because the delta inside the root is negative. Therefore, this equation never equals zero, it hovers above the X-axis, and it has no zero points.

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Question 1

The following function has been graphed below:

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

Calculate points A and B.

Question 2

The following function has been graphed below:

$$ f(x)=x^2-6x+5 $$

Calculate points A and B.

$$

Question 3

The following function has been graphed below:

$$ f(x)=x^2-6x+8 $$

Calculate points A and B.

Examples with solutions for Zeros of a Fuction

Exercise #1

The following function has been graphed below:

$f(x)=x^2-8x+16$

Calculate point C.

Video Solution

Step-by-Step Solution

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

Answer

$(4,0)$

Exercise #2

Determine the points of intersection of the function

$y=(x-5)(x+5)$

With the X

Video Solution

Step-by-Step Solution

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!

Answer

$(5,0),(-5,0)$

Exercise #3

The following function has been graphed below:

$f(x)=-x^2+5x+6$

Calculate points A and B.

Video Solution

Answer

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

Exercise #4

The following function has been graphed below:

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

Calculate points A and B.

Video Solution

Answer

$A(-1,0),B(4,0)$

Exercise #5

The following function has been graphed below:

$f(x)=x^2-6x+5$

Calculate points A and B.

Video Solution

Answer

$(1,0),(5,0)$

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Finding the Zeros of a Parabola

Finding the zeros of a quadratic function of the form \(y=ax^2+bx+c\)

When trying to find the zero point, you can encounter three possible results:

Test yourself on zeros of a fuction!

Examples with solutions for Zeros of a Fuction

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Answer

Exercise #4

Video Solution

Answer

Exercise #5

Video Solution

Answer

More Questions

Zeros of a Fuction