Match the Quadratic Function y = x² + 9 to Its Correct Graph

Q: How do I know which direction the parabola opens?

Look at the coefficient of $ x^2 $! Since it's positive (+1), the parabola opens upward. If it were negative, it would open downward.

Q: What does the +9 actually do to the graph?

The +9 is a vertical shift. It moves every point on the basic parabola $ y = x^2 $ up by 9 units, so the vertex moves from (0,0) to (0,9).

Q: How can I find the y-intercept quickly?

Set x = 0 in the equation: $ y = 0^2 + 9 = 9 $. The y-intercept is always at (0, 9), which is also the vertex for this function!

Q: Why is the vertex at (0, 9) and not (0, -9)?

The vertex is the lowest point since the parabola opens upward. With $ y = x^2 + 9 $, the minimum y-value is 9 (when x = 0), making the vertex (0, 9).

Q: How do I distinguish between similar-looking graphs?

Check the vertex location and y-intercept. For $ y = x^2 + 9 $, both are at (0, 9). Look for the graph where the lowest point is 9 units above the x-axis.

Quadratic Functions with Vertical Shifts

One function

$y=x^2+9$

to the corresponding graph:

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Match the function to the appropriate graph

00:03 Notice the coefficient of X squared is positive, so the function is 'happy'

00:08 We want to find the intersection point with the Y-axis

00:12 Substitute X=0 and solve to find the intersection point with Y-axis

00:19 This is the intersection point with the Y-axis

00:22 According to the function type and intersection point

00:26 We can conclude there are no intersection points with the X-axis

00:31 Let's draw the graph according to the function type and intersection points we found

00:34 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

One function

$y=x^2+9$

to the corresponding graph:

Final Answer

Key Points to Remember

Essential concepts to master this topic

Vertex Form: $y = x^2 + 9$ has vertex at (0, 9)
Y-intercept: When x = 0, y = 9 locates starting point
Verification: Graph opens upward with lowest point at y = 9 ✓

Common Mistakes

Avoid these frequent errors

Confusing the direction of the vertical shift
Don't think +9 moves the graph down = parabola below x-axis! Adding 9 shifts the basic parabola UP 9 units from the origin. Always remember: +k moves UP, -k moves DOWN in $y = x^2 + k$ .

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 which direction the parabola opens?

Look at the coefficient of $x^2$ ! Since it's positive (+1), the parabola opens upward. If it were negative, it would open downward.

What does the +9 actually do to the graph?

The +9 is a vertical shift. It moves every point on the basic parabola $y = x^2$ up by 9 units, so the vertex moves from (0,0) to (0,9).

How can I find the y-intercept quickly?

Set x = 0 in the equation: $y = 0^2 + 9 = 9$ . The y-intercept is always at (0, 9), which is also the vertex for this function!

Why is the vertex at (0, 9) and not (0, -9)?

The vertex is the lowest point since the parabola opens upward. With $y = x^2 + 9$ , the minimum y-value is 9 (when x = 0), making the vertex (0, 9).

How do I distinguish between similar-looking graphs?

Check the vertex location and y-intercept. For $y = x^2 + 9$ , both are at (0, 9). Look for the graph where the lowest point is 9 units above the x-axis.

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