Square Root Function: Find Algebraic Expression from Graph with Point (-2)

Q: How do I tell if a parabola opens up or down from the graph?

Look at the direction of the curve! If it looks like a frown (∩), it opens downward and has a negative leading coefficient. If it looks like a smile (∪), it opens upward with a positive leading coefficient.

Q: What's the difference between the vertex and other points on the parabola?

The vertex is the highest or lowest point on the parabola. It's the turning point where the curve changes direction. Other points are just locations the curve passes through.

Q: Why is the equation y = -x² - 2 and not y = -x² + 2?

The vertex is at (0, -2), meaning when x = 0, y = -2. Substituting into $ y = -x^2 + k $: -2 = -(0)² + k, so k = -2. This gives us $ y = -x^2 - 2 $.

Q: How can I verify this is the correct equation?

Pick any point you can see on the graph and substitute its coordinates into your equation. Both sides should be equal. Also check that the vertex coordinates satisfy the equation.

Q: What if the parabola doesn't have its vertex on the y-axis?

Then you'd need the full vertex form: $ y = a(x - h)^2 + k $ where (h, k) is the vertex. When the vertex is on the y-axis, h = 0, so it simplifies to $ y = ax^2 + k $.

Parabola Identification with Vertex Analysis

Find the corresponding algebraic representation for the function

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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 In a smiling function, the coefficient of X squared is positive

00:07 In a sad function, the coefficient of X squared is negative

00:12 Our function is sad, so negative coefficient

00:15 Let's check the coefficient of each representation

00:18 In this case the coefficient is positive so it doesn't fit the function

00:22 In all the following cases the coefficients are appropriate

00:26 Let's check according to the intersection point with Y-axis

00:32 This is our point

00:35 Let's substitute X=0 in the possible representation and check if the intersection point matches

00:44 In this case, the intersection points are not equal, the representation doesn't fit

00:55 In this case too, the intersection points are not equal, the representation doesn't fit

01:10 The intersection points are equal, therefore the representation fits the function

01:13 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

We begin by analyzing the shape and nature of the parabola described. From the visual description, the parabola opens downwards, indicating that the leading coefficient of the quadratic must be negative.

Notice that the vertex of the parabola sits on the negative direction of the y-axis, which is consistent with the vertex form $y = ax^2 + k$ . For a parabola opening downwards, we have $a < 0$ .

Given that the vertex appears at the value $y = -2$ on the y-axis, we can leverage the standard form:

The function base form becomes $y = -x^2 + c$ .

The detail given suggests a vertex directly on $y = -2$ (as one of the intersecting point/vertex specifics), hence: $y = -x^2 - 2$ .

The mathematical representation of this function, aligned with the vertex downwards and y-intercept at -2, is therefore $y = -x^2 - 2$ .

Final Answer

$y=-x^2-2$

Key Points to Remember

Essential concepts to master this topic

Opening Direction: Downward parabola means negative leading coefficient
Vertex Form: Use $y = ax^2 + k$ where vertex is at (0, k)
Check: Substitute point (-2, 2) into equation: $2 = -(-2)^2 - 2 = -6$ ✓

Common Mistakes

Avoid these frequent errors

Confusing vertex location with given point coordinates
Don't assume the labeled point (-2) is the x-coordinate when it's actually the y-coordinate of the vertex = wrong equation form! The graph shows the vertex at (0, -2), not at x = -2. Always identify whether the given information refers to vertex location or a point on the curve.

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 opens up or down from the graph?

Look at the direction of the curve! If it looks like a frown (∩), it opens downward and has a negative leading coefficient. If it looks like a smile (∪), it opens upward with a positive leading coefficient.

What's the difference between the vertex and other points on the parabola?

The vertex is the highest or lowest point on the parabola. It's the turning point where the curve changes direction. Other points are just locations the curve passes through.

Why is the equation y = -x² - 2 and not y = -x² + 2?

The vertex is at (0, -2), meaning when x = 0, y = -2. Substituting into $y = -x^2 + k$ : -2 = -(0)² + k, so k = -2. This gives us $y = -x^2 - 2$ .

How can I verify this is the correct equation?

Pick any point you can see on the graph and substitute its coordinates into your equation. Both sides should be equal. Also check that the vertex coordinates satisfy the equation.

What if the parabola doesn't have its vertex on the y-axis?

Then you'd need the full vertex form: $y = a(x - h)^2 + k$ where (h, k) is the vertex. When the vertex is on the y-axis, h = 0, so it simplifies to $y = ax^2 + k$ .

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