Quadratic Function: Converting Graph to Algebraic Form at Point (-2)

Q: How do I know this parabola opens upward?

The parabola in the graph opens upward, which means the coefficient of $ x^2 $ is positive. Since it's a standard parabola shape, the coefficient is 1, giving us $ y = x^2 + c $.

Q: What if the parabola doesn't cross the y-axis on my graph?

Every parabola of the form $ y = x^2 + c $ crosses the y-axis! Look carefully at where x = 0 on your graph. The y-coordinate at that point is your constant term.

Q: Why isn't the answer just y = x²?

Because $ y = x^2 $ would cross the y-axis at (0,0). Our graph shows the parabola crosses at (0,-2), so we need the constant term -2 to shift it down.

Q: How can I tell the difference between y = x² - 2 and y = -x²?

Look at the direction the parabola opens! $ y = x^2 - 2 $ opens upward (U-shape), while $ y = -x^2 $ opens downward (∩-shape). Our graph opens upward.

Quadratic Functions with Y-Intercept Identification

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

00:00 Choose the appropriate algebraic representation for the function

00:03 Find the intersection point of the function with the Y-axis

00:10 This is the point

00:14 Let's substitute X=0 in each function and see if it matches

00:18 This option is not suitable

00:25 Continue with this method for each representation and check the intersection point

00:33 This option is also not suitable

00:44 This option is suitable

00:55 This option is not suitable

01:02 And this is the solution to the question

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

The problem involves determining the algebraic formula of a function represented by a graph of a parabola. From the axes on the graph, we observe that the graph intersects the y-axis at $y = -2$ . This intersection indicates that the constant term $c$ in the quadratic function $y = x^2 + c$ is $-2$ .

Step 1: Start with the parabola formula $y = x^2 + c$ . Since this is a standard vertical parabola centered on the y-axis (no $x$ term with a coefficient), it takes the form $y = ax^2 + c$ .
Step 2: From the graph, we see that when $x = 0$ , $y = -2$ . Substituting into the equation gives us: $y = 0^2 + c = -2$ .
Step 3: Solve for $c$ . Here, it is evident that $c = -2$ since the equation simplifies directly to this when $x = 0$ .

Therefore, the equation of the parabola is $y = x^2 - 2$ , which corresponds to choice 3 in the provided options.

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

Final Answer

$y=x^2-2$

Key Points to Remember

Essential concepts to master this topic

Y-Intercept Rule: Where parabola crosses y-axis determines constant term
Technique: At x = 0, equation becomes y = 0² + c = c
Check: Substitute x = 0 into $y = x^2 - 2$ gives y = -2 ✓

Common Mistakes

Avoid these frequent errors

Confusing vertex with y-intercept
Don't look at the vertex point to find the constant term = wrong equation! The vertex tells you the parabola's turning point, not where it crosses the y-axis. Always find where the graph crosses the y-axis (when x = 0) to determine the constant term.

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 know this parabola opens upward?

The parabola in the graph opens upward, which means the coefficient of $x^2$ is positive. Since it's a standard parabola shape, the coefficient is 1, giving us $y = x^2 + c$ .

What if the parabola doesn't cross the y-axis on my graph?

Every parabola of the form $y = x^2 + c$ crosses the y-axis! Look carefully at where x = 0 on your graph. The y-coordinate at that point is your constant term.

Why isn't the answer just y = x²?

Because $y = x^2$ would cross the y-axis at (0,0). Our graph shows the parabola crosses at (0,-2), so we need the constant term -2 to shift it down.

How can I tell the difference between y = x² - 2 and y = -x²?

Look at the direction the parabola opens! $y = x^2 - 2$ opens upward (U-shape), while $y = -x^2$ opens downward (∩-shape). Our graph opens upward.

What does the -2 point marked on the graph represent?

The point marked as -2 shows where the parabola crosses the y-axis. This y-intercept value becomes the constant term in your equation, giving us $y = x^2 - 2$ .

🌟 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