Match y=-2x²-3 to its Corresponding Parabola Graph

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

Look at the coefficient of $ x^2 $! In $ y = -2x^2 - 3 $, the coefficient is -2 (negative), so it opens downward. Positive coefficient = opens upward.

Q: Why is the vertex at (0, -3)?

Since there's no $ x $ term in $ y = -2x^2 - 3 $, the parabola is symmetric about the y-axis. The vertex occurs at x = 0, giving us the point (0, -3).

Q: What if two graphs look similar?

Check specific points! Try $ x = 1 $: $ y = -2(1)^2 - 3 = -5 $. The correct graph should pass through (1, -5) and (-1, -5) due to symmetry.

Quadratic Functions with Downward-Opening Parabolas

One function

$y=-2x^2-3$

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 negative, so the function is concave down

00:11 We want to find the Y-axis intersection point

00:14 Substitute X=0 and solve to find the Y-axis intersection point

00:20 This is the Y-axis intersection point

00:24 Let's draw the graph according to the function type and the intersection point we found

00:30 And this is the solution to the question

Step-by-step written solution

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Understand the problem

One function

$y=-2x^2-3$

to the corresponding graph:

Step-by-step solution

To solve this problem, we'll match the given function $y = -2x^2 - 3$ with its corresponding graph based on specific characteristics:

The function $y = -2x^2 - 3$ is a quadratic equation representing a parabola.
Since the coefficient of $x^2$ is negative, the parabola opens downward.
The y-intercept is -3, which means the parabola crosses the y-axis at $-3$ .
The maximum point (vertex) of the parabola occurs at its axis of symmetry, from which we know it opens downward from that point.

Given these observations, we analyze each graphical option:

Graph 1 represents a parabola opening upward, so it does not match.
Graph 2 might have an appropriate direction but not the correct intercept.
Graph 3 doesn't match key features such as y-intercept and direction.
Graph 4 shows a downward opening parabola with its intercept significantly influenced by negative vertical shift, which matches $y = -2x^2 - 3$ .

Therefore, the function $y = -2x^2 - 3$ matches with graph option 4.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Direction Rule: Negative coefficient of $x^2$ means parabola opens downward
Y-Intercept: Set x = 0 to find y = -3 where graph crosses y-axis
Vertex Check: Maximum point at (0, -3) since parabola opens down ✓

Common Mistakes

Avoid these frequent errors

Confusing upward and downward opening parabolas
Don't assume all parabolas open upward = wrong graph selection! The sign of the $x^2$ coefficient determines direction. Always check: negative coefficient means opens downward, positive means opens upward.

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 way the parabola opens?

Look at the coefficient of $x^2$ ! In $y = -2x^2 - 3$ , the coefficient is -2 (negative), so it opens downward. Positive coefficient = opens upward.

What does the -3 tell me about the graph?

The -3 is the y-intercept! It tells you where the parabola crosses the y-axis. Since there's no x term, the vertex is also at (0, -3).

Why is the vertex at (0, -3)?

Since there's no $x$ term in $y = -2x^2 - 3$ , the parabola is symmetric about the y-axis. The vertex occurs at x = 0, giving us the point (0, -3).

How do I eliminate wrong graph choices quickly?

Use these two quick checks: 1) Does it open the right direction? 2) Does it cross the y-axis at the right point? Any graph failing either test is automatically wrong!

What if two graphs look similar?

Check specific points! Try $x = 1$ : $y = -2(1)^2 - 3 = -5$ . The correct graph should pass through (1, -5) and (-1, -5) due to symmetry.

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