Identify the Graph of y=x²-9: Quadratic Function Recognition

Q: How do I find the vertex of y = x² - 9?

For the form $ y = x^2 + k $, the vertex is always at (0, k). Since k = -9, the vertex is at (0, -9). This is the lowest point of the parabola.

Q: Why does the parabola open upward?

The coefficient of $ x^2 $ is positive (it's +1). When this coefficient is positive, the parabola opens upward like a U-shape. If it were negative, it would open downward.

Q: What's the difference between y = x² and y = x² - 9?

$ y = x^2 $ has its vertex at (0, 0), while $ y = x^2 - 9 $ shifts this down by 9 units to (0, -9). The shape stays the same, just moved vertically!

Q: How can I check if I picked the right graph?

Look for these key features: Vertex at (0, -9)Parabola opens upwardY-intercept at -9Symmetric about the y-axis

Q: What happens when x = 3 or x = -3?

When x = ±3: $ y = (±3)^2 - 9 = 9 - 9 = 0 $. These are the x-intercepts where the parabola crosses the x-axis at (-3, 0) and (3, 0).

Parabola Vertex with Y-intercept Translation

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

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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, therefore the function is smiling

00:09 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

00:17 This is the intersection point with the Y-axis

00:20 Now let's find the intersection point with the X-axis

00:23 Substitute Y=0 to find the intersection point with the X-axis

00:27 Isolate X

00:30 Take the square root

00:35 When taking square root there are 2 solutions (positive and negative)

00:40 These are the intersection points with the X-axis

00:46 Let's draw the graph according to the function type and intersection points we found

00:57 And this is the solution to the question

Step-by-step written solution

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Understand the problem

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

Step-by-step solution

To solve the problem of identifying which chart represents the function $y = x^2 - 9$ , let's analyze the function and its graph:

The function $y = x^2 - 9$ is a parabola that can be described by the general form $y = x^2 + k$ where $k = -9$ .
It is a standard upward-opening parabola with its vertex located at the point $(0, -9)$ . This is because there is no coefficient affecting $x$ , so horizontally it is centered at the origin.
To find the correct graph, we look for one where the bottommost point of the parabola is at $(0, -9)$ . This point, known as the vertex, should sit on the y-axis and be the lowest point of the curve due to the upward opening.

After inspecting the charts:

Chart 4 depicts a parabola opening upwards, with its vertex at $(0, -9)$ . This aligns perfectly with the form and properties of our function $y = x^2 - 9$ .

Therefore, the chart that represents the function $y = x^2 - 9$ is Choice 4.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Vertex Form: For $y = x^2 - 9$ , vertex is at (0, -9)
Y-intercept: Set x = 0 to get y = -9 on axis
Check: Vertex should be lowest point on upward parabola ✓

Common Mistakes

Avoid these frequent errors

Confusing vertex location with y-intercept
Don't think the vertex is at (0, 9) instead of (0, -9) = graph appears shifted up! Students often forget the negative sign affects the vertical position. Always remember that $y = x^2 - 9$ shifts the basic parabola DOWN by 9 units.

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 find the vertex of y = x² - 9?

For the form $y = x^2 + k$ , the vertex is always at (0, k). Since k = -9, the vertex is at (0, -9). This is the lowest point of the parabola.

Why does the parabola open upward?

The coefficient of $x^2$ is positive (it's +1). When this coefficient is positive, the parabola opens upward like a U-shape. If it were negative, it would open downward.

What's the difference between y = x² and y = x² - 9?

$y = x^2$ has its vertex at (0, 0), while $y = x^2 - 9$ shifts this down by 9 units to (0, -9). The shape stays the same, just moved vertically!

How can I check if I picked the right graph?

Look for these key features:

Vertex at (0, -9)
Parabola opens upward
Y-intercept at -9
Symmetric about the y-axis

What happens when x = 3 or x = -3?

When x = ±3: $y = (±3)^2 - 9 = 9 - 9 = 0$ . These are the x-intercepts where the parabola crosses the x-axis at (-3, 0) and (3, 0).

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