Match the Quadratic Function y = 6x² to its Corresponding Graph

Q: How do I know if a parabola opens up or down?

Look at the coefficient of $ x^2 $! If it's positive (like +6), the parabola opens upward. If it's negative (like -6), it opens downward.

Q: Why does a larger coefficient make the parabola narrower?

Think of it as vertical stretching! When $ a = 6 $, every y-value is 6 times taller than the basic parabola $ y = x^2 $. This pulls the sides up faster, making it look narrower.

Q: How can I tell which graph is narrower?

Compare how quickly the parabolas rise! A narrow parabola shoots up steeply near the vertex, while a wide parabola rises more gradually. Graph 2 rises much faster than graph 4.

Q: What if the coefficient was between 0 and 1?

If $ 0 (like \( a = 0.5 $), the parabola would be wider than $ y = x^2 $ because it's vertically compressed instead of stretched.

Q: Where is the vertex of this parabola?

The vertex is at (0, 0) because there are no additional terms. The function $ y = 6x^2 $ has no horizontal or vertical shifts from the origin.

Quadratic Functions with Vertical Stretch Coefficients

One function

$y=6x^2$

to the corresponding graph:

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's match the function to the correct graph.

00:10 Notice the X squared term has a positive coefficient. So, the graph opens upward, like a smile.

00:17 Next, we'll find where the graph crosses the Y-axis.

00:21 Set X to zero, then solve to find this intersection point.

00:26 This point is where the graph meets both the Y-axis, and often the X-axis.

00:32 Now, draw the graph using the function's shape and our intersection point.

00:37 And there you have it! That's how we solve the problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

One function

$y=6x^2$

to the corresponding graph:

Step-by-step solution

The function given is $y = 6x^2$ . This is a quadratic function, a type of parabola with vertex at the origin (0,0), because there are no additional terms indicating a horizontal or vertical shift.

First, note the coefficient of $x^2$ is $6$ . A positive coefficient indicates that the parabola opens upwards. The value of $6$ means the parabola is relatively narrow, as it is stretched vertically compared to the standard $y = x^2$ .

To identify the corresponding graph:

Recognize that a function of the form $y = ax^2$ with $a > 1$ indicates a narrower parabola.
Out of the given graphs, we should look for an upward-opening narrow parabola.

Upon examining each graph, you find that option 2 shows a parabola that is narrower than the standard parabola $y = x^2$ and opens upwards distinctly, matching our function $y = 6x^2$ .

Therefore, the correct graph for the function $y = 6x^2$ is option 2.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Form: $y = ax^2$ with $a > 0$ opens upward from vertex (0,0)
Stretch Factor: $a = 6$ makes parabola 6 times narrower than $y = x^2$
Check: Test point (1,6): $y = 6(1)^2 = 6$ matches graph ✓

Common Mistakes

Avoid these frequent errors

Confusing narrow vs wide parabolas
Don't think larger coefficients make wider parabolas = choosing graph 4 instead of 2! When $a > 1$ , the parabola stretches vertically and appears narrower. Always remember: larger coefficient = narrower parabola.

Practice Quiz

Test your knowledge with interactive questions

Find the ascending area of the function

$$ f(x)=2x^2 $$

FAQ

Everything you need to know about this question

How do I know if a parabola opens up or down?

Look at the coefficient of $x^2$ ! If it's positive (like +6), the parabola opens upward. If it's negative (like -6), it opens downward.

Why does a larger coefficient make the parabola narrower?

Think of it as vertical stretching! When $a = 6$ , every y-value is 6 times taller than the basic parabola $y = x^2$ . This pulls the sides up faster, making it look narrower.

How can I tell which graph is narrower?

Compare how quickly the parabolas rise! A narrow parabola shoots up steeply near the vertex, while a wide parabola rises more gradually. Graph 2 rises much faster than graph 4.

What if the coefficient was between 0 and 1?

If $0 < a < 1$ (like $a = 0.5$ ), the parabola would be wider than $y = x^2$ because it's vertically compressed instead of stretched.

Where is the vertex of this parabola?

The vertex is at (0, 0) because there are no additional terms. The function $y = 6x^2$ has no horizontal or vertical shifts from the origin.

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