Find the Algebraic Equation: Parabola Passing Through (4,0)

Q: How do I tell if a parabola is shifted up or down?

Look at where the parabola crosses the y-axis! If it crosses above the origin, it's shifted up. If below, it's shifted down. The number tells you how many units.

Q: What does the horizontal line labeled '4' mean in the graph?

The horizontal line at y = 4 shows the y-intercept of the parabola. This means when x = 0, y = 4, indicating the parabola is shifted 4 units upward from the basic $ y = x^2 $.

Q: Why isn't the answer just y = x²?

The basic parabola $ y = x^2 $ passes through (0,0). But this graph shows the parabola passing through (0,4), so we need to add 4 to shift it upward.

Q: What if the parabola was shifted down instead?

If the parabola crossed the y-axis below the origin, you'd subtract instead of add. For example, crossing at (0,-3) would give $ y = x^2 - 3 $.

Quadratic Functions with Vertical Shifts

Find the corresponding algebraic representation for the function

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the corresponding algebraic representation for the function

Step-by-step solution

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

Identify the graphical points where the parabola interacts with the axes.
Determine the role of any numbers indicated along the axes, specifically for vertical shifts.
Write the function based on these observations.

Now, let's work through each step:
Step 1: The parabola is shown as an intersection with a horizontal line labeled "4". This suggests that the parabola is shifted upwards by 4 units.
Step 2: The standard quadratic equation without shifts is $y = x^2$ . Therefore, to account for a shift of 4 units upwards, we modify this to $y = x^2 + 4$ .
Step 3: With this shift identified, the algebraic representation of the function is completed.

Therefore, the solution to the problem is $y = x^2 + 4$ .

Final Answer

$y=x^2+4$

Key Points to Remember

Essential concepts to master this topic

Vertical Shift: Adding constants moves parabola up or down
Technique: Transform $y = x^2$ to $y = x^2 + 4$ for upward shift
Check: Verify y-intercept: when x = 0, y = 4 ✓

Common Mistakes

Avoid these frequent errors

Confusing x-intercepts with vertical shifts
Don't assume the point (4,0) means the parabola passes through it = wrong equation! The graph shows a horizontal line at y = 4, indicating a vertical shift. Always identify whether marked points show intercepts or shift indicators.

Practice Quiz

Test your knowledge with interactive questions

Which chart represents the function $$ y=x^2-9 $$ ?

FAQ

Everything you need to know about this question

How do I tell if a parabola is shifted up or down?

Look at where the parabola crosses the y-axis! If it crosses above the origin, it's shifted up. If below, it's shifted down. The number tells you how many units.

What does the horizontal line labeled '4' mean in the graph?

The horizontal line at y = 4 shows the y-intercept of the parabola. This means when x = 0, y = 4, indicating the parabola is shifted 4 units upward from the basic $y = x^2$ .

Why isn't the answer just y = x²?

The basic parabola $y = x^2$ passes through (0,0). But this graph shows the parabola passing through (0,4), so we need to add 4 to shift it upward.

How can I verify my equation is correct?

Test the y-intercept! Substitute x = 0 into your equation. For $y = x^2 + 4$ : when x = 0, y = 0 + 4 = 4. This matches the graph! ✓

What if the parabola was shifted down instead?

If the parabola crossed the y-axis below the origin, you'd subtract instead of add. For example, crossing at (0,-3) would give $y = x^2 - 3$ .

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Parabola Families questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime