18.1 How to Find the y-Intercept of a Quadratic Graph

🟦 Concept

The y-intercept is the point where a quadratic curve crosses the y-axis.

On the y-axis:$$x=0$$

A quadratic equation is usually written as:$$y=ax^2+bx+c$$

When we substitute $x=0$, we get:$$y=a(0)^2+b(0)+c$$$$y=c$$

Therefore, the y-intercept is:$$(0,c)$$

Easy Rule

In $y=ax^2+bx+c$, the value of $c$ gives the y-intercept.

---

🟩 Example 1

Consider:$$y=x^2-4x+3$$

Here:$$a=1,\qquad b=-4,\qquad c=3$$

Since:$$c=3$$

the y-intercept is:$$(0,3)$$

You can also check this by substituting:$$x=0$$$$y=(0)^2-4(0)+3$$$$y=3$$

Therefore:$$\text{y-intercept}=(0,3)$$

---

🟪 Example 2

Consider:$$y=2x^2+5x-7$$

Here:$$c=-7$$

Therefore, the y-intercept is:$$(0,-7)$$

This means the curve crosses the y-axis below the x-axis.

---

🟨 What If $c=0$?

Consider:$$y=x^2-4x$$

Here:$$c=0$$

So the y-intercept is:$$(0,0)$$

This means the quadratic curve passes through the origin.

---

💡 Exam Tip

If the quadratic equation is already written as:$$y=ax^2+bx+c$$

you normally do not need a long calculation.

Simply identify $c$.

Then write:$$\text{y-intercept}=(0,c)$$

---

⚠️ Common Mistake

Do not write only:$$3$$

if the question asks for the coordinates of the y-intercept.

For:$$y=x^2-4x+3$$

the correct coordinate is:$$(0,3)$$

The first coordinate must be $0$ because every point on the y-axis has:$$x=0$$

---

❓ Quick Check

For:$$y=3x^2-2x+5$$

what is the y-intercept?

Since:$$c=5$$

the y-intercept is:$$(0,5)$$

For:$$y=-2x^2+7x-4$$

the y-intercept is:$$(0,-4)$$

---

⭐ Quick Summary

For a quadratic equation:$$y=ax^2+bx+c$$

the y-intercept occurs when:$$x=0$$

Therefore:$$y=c$$

So:$$\text{y-intercept}=(0,c)$$

Remember

cc gives the y-intercept. c⟶(0,c)