Table of Contents

Last modified on October 9th, 2024

The **x-intercepts** of a parabola are the points where the U-shaped curve intersects the x-axis. A parabola can have 0, 1, or 2 x-intercepts, as shown.

If a parabola has 1 x-intercept, the x-axis is said to be the tangent to the parabola.

At any x-intercept, the value of y is always 0. The x and y values can be obtained by solving the quadratic function y = ax^{2} + bx + c, where the discriminant b^{2} – 4ac determines the nature of the roots.

To find the number of x-intercepts for the general quadratic function y = ax^{2} + bx + c:

- If the discriminant b
^{2}– 4ac < 0, the number of x-intercepts is 0 - If the discriminant b
^{2}– 4ac = 0, the number of x-intercepts is 1 - If the discriminant b
^{2}– 4ac > 0, the number of x-intercepts is 2

Let us find the number of x-intercepts of the parabola y = 2x^{2} + 7x + 5

Given, y = 2x^{2} + 7x + 5 …..(i)

Comparing the equation (i) of the parabola with the standard form y = ax^{2} + bx + c, we have

a = 2, b = 7, and c = 5

Now, the discriminant is b^{2} – 4ac = 7^{2} – 4 × 2 × 5 = 9 > 0

Here, b^{2} – 4ac > 0, and thus, the number of x-intercepts is 2

The x-intercepts of a parabola determine the roots of its equation and identify the points where the parabola intersects the x-axis. To find the x-intercept(s), we follow the following steps:

- Setting y = 0 in the equation of the parabola
- Solving for x either by factoring or quadratic formula: ${x=\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}}$
- Writing the roots in the coordinate form (x, 0)

Let us find the x-intercepts of the parabola y = 2x^{2} + 7x + 5** **

2x^{2} + 7x + 5 = 0

Here, we will solve by factoring

⇒ 2x^{2} + 5x + 2x + 5 = 0

⇒ x(2x + 5) + (2x + 5) = 0

⇒ (2x + 5)(x + 1) = 0

Here, (x + 1) = 0 ⇒ x = -1

(2x + 5) = 0 ⇒ x = ${-\dfrac{5}{2}}$ = -2.5

Thus, the x-intercepts (roots) are (-2.5, 0) and (-1, 0)

Plotting the given parabola in the graph, we get the x-intercepts, as shown.

**Find the x-intercepts of the parabola represented by the equation: y = x ^{2} – 4x – 5**

Solution:

Given, y = x^{2} – 4x – 5 …..(i)

Considering y = 0 in the equation (i), we get

x^{2} – 4x – 5 = 0

⇒ x^{2} – 5x + x – 5 = 0

⇒ x(x – 5) + (x – 5) = 0

⇒ (x + 1)(x – 5) = 0

Now, (x + 1) = 0 ⇒ x = -1

(x – 5) = 0 ⇒ x = 5

Thus, the x-intercepts are (-1, 0) and (5, 0)

**Determine the x-intercepts of the parabola y = 2x ^{2} – 6x + 4 using the quadratic root formula**

Solution:

Given, y = 2x^{2} – 6x + 4 …..(i)

As we know, the standard form of the parabola is y = ax^{2} + bx + c …..(ii)

Comparing the equations (i) and (ii), we have

a = 2, b = -6, c = 4

Now, setting y = 0, we get

2x^{2} – 6x + 4 = 0

⇒ ${x=\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}}$

⇒ ${x=\dfrac{-\left( -6\right) \pm \sqrt{\left( -6\right) ^{2}-4\times 2\times 4}}{2\times 2}}$

⇒ x = 2 and x = 1

Thus, the x-intercepts are (2, 0) and (1, 0)

**Problem – **Finding the x-intercept of a parabola with a **SINGLE INTERCEPT**

**Find the x-intercepts of the parabola y = x ^{2} + 2x + 1**

Solution:

Given, y = x^{2} + 2x + 1 …..(i)

As we know, the standard form of the parabola is y = ax^{2} + bx + c …..(ii)

Comparing the equations (i) and (ii), we have

a = 1, b = 2, and c = 1

Here, the discriminant is b^{2} – 4ac = (2)^{2} – 4(1)(1) = 0

Thus, the parabola has 1 x-intercept.

Considering y = 0, we get

x^{2} + 2x + 1 = 0

⇒ (x + 1)^{2} = 0

⇒ (x + 1) = 0

⇒ x = -1

Thus, the x-intercept is (-1, 0)

Like x-intercepts, each parabola contains a **y-intercept**, the point at which the parabola intercepts the y-axis.

To find the y-intercept, we follow the following steps:

- Setting x = 0 in the equation
- Solving for y
- Writing y in the coordinate form (0, y)

Now, let us find the y-intercepts of the parabola y = 2x^{2} + 7x + 5** **

y = 2(0)^{2} + 7(0) + 5

⇒ y = 5

Thus, the y-intercept is (0, 5)

Plotting the given parabola on the graph, we get the y-intercept, as shown.

**Determine the x and y-intercepts of the following parabola:****y = 2x ^{2} + 3x – 5**

Solution:

Given, y = 2x^{2} + 3x – 5 …..(i)

Considering y = 0 in the equation (i), we get

2x^{2} + 3x – 5 = 0

⇒ 2x^{2} + 5x – 2x – 5 = 0

⇒ x(2x + 5) – (2x + 5) = 0

⇒ (2x + 5)(x – 1) = 0

Now, (2x + 5) = 0 ⇒ x = ${-\dfrac{5}{2}}$ = -2.5

(x – 1) = 0 ⇒ x = 1

Thus, the x-intercepts are (-2.5, 0) and (1, 0)

By putting x = 0 in the equation (i), we get

y = 2(0)^{2} + 3(0) – 5

⇒ y = -5

Thus, the y-intercept is (0, -5)

**Find the x and y-intercepts of y = 4x ^{2} – 8x**

Solution:

Given, y = 4x^{2} – 8x …..(i)

Considering y = 0 in the equation (i), we get

4x^{2} – 8x = 0

⇒ 4x(x – 2) = 0

Now, 4x = 0 ⇒ x = 0

(x – 2) = 0 ⇒ x = 2

Thus, the x-intercepts are (0, 0) and (2, 0)

By putting x= 0 in the equation (i), we get

y = 4(0)^{2} – 8(0)

⇒ y = 0

Thus, the y-intercept is (0, 0)

**Find the x and y-intercepts of the parabola y = 5x ^{2} + 15x + 10**

Solution:

Given, y = 5x^{2} + 15x + 10 …..(i)

Considering y = 0 in the equation (i), we get

5x^{2} + 15x + 10 = 0

⇒ x^{2} + 3x + 2 = 0

⇒ x^{2} + 2x + x + 2 = 0

⇒ x(x + 2) + (x + 2) = 0

⇒ (x + 2)(x + 1) = 0

Now, (x + 2) = 0 ⇒ x = -2

(x + 1) = 0 ⇒ x = -1

Thus, the x-intercepts are (-2, 0) and (-1, 0)

By putting x = 0 in the equation (i), we get

y = 5(0)^{2} + 15(0) + 10

⇒ y = 10

Thus, the y-intercept is (10, 0)

**Problem –** Finding **X & Y INTERCEPTS** of a parabola given in **VERTEX FORM**

**Determine the x and y intercepts of the parabola y = 2(x – 2) ^{2} – 3**

Solution:

Given, y = 2(x – 2)^{2} – 3 …..(i)

Considering y = 0 in the equation (i), we get

2(x – 2)^{2} – 3 = 0

⇒ 2(x^{2} – 4x + 4) – 3 = 0

⇒ 2x^{2} – 8x + 8 – 3 = 0

⇒ 2x^{2} – 8x + 5 = 0 …..(ii)

As we know, the standard form of the parabola is y = ax^{2} + bx + c …..(iii)

Comparing the equations (ii) and (iii), we have

a = 2, b = -8, c = 5

Here, ${x=\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}}$

⇒ ${x=\dfrac{-\left( -8\right) \pm \sqrt{\left( -8\right) ^{2}-4\times 2\times 5}}{2\times 2}}$

⇒ x = ${2+\sqrt{\dfrac{3}{2}}}$ and x = ${2-\sqrt{\dfrac{3}{2}}}$

Thus, the x-intercepts are ${\left( 2+\sqrt{\dfrac{3}{2}},0\right)}$ and ${\left( 2-\sqrt{\dfrac{3}{2}},0\right)}$

By putting x = 0 in the equation (i), we get

y = 2(0 – 2)^{2} – 3 = 8 – 3 = 5

⇒ y = 5

Thus, the y-intercept is (0, 5)

Last modified on October 9th, 2024