Find the Quadratic Equation Passing Through Point (4,6)

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

Look at the direction of the curve in the graph! If it opens upward like a U-shape, then $a > 0$. If it opens downward like an upside-down U, then \(a .

Q: What if I can't see the vertex clearly on the graph?

The vertex is the highest or lowest point of the parabola. In this problem, it's labeled as (4,6), which becomes your $h$ and $k$ values.

Q: Why is the vertex form better than standard form here?

Vertex form $ y = a(x-h)^2 + k $ directly shows you the vertex coordinates and makes it easy to write the equation when you know the vertex.

Q: How do I find the value of 'a' if it's not given?

Use another point on the parabola! Substitute the coordinates into your equation and solve for $a$. In this case, the graph's direction tells us $a = -1$.

Q: What's the difference between (x-4) and (x+4) in the equation?

The sign is opposite of the x-coordinate! If the vertex is at $x = 4$, you write (x-4). If it were at $x = -4$, you'd write (x+4).

Vertex Form with Parabola Direction

Choose the equation that represents the following:

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the appropriate algebraic representation for the function

00:05 The function is a parabola

00:27 The function is sad (opens downward), therefore coefficient A is negative

00:39 The maximum point is 4 units to the right according to the graph, that's how we find P

01:11 The maximum point is 6 units up according to the graph, that's how we find K

01:33 And that's the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the equation that represents the following:

Step-by-step solution

To determine the correct equation, we need to consider the vertex form of a parabola:

The vertex form is given by $y = a(x-h)^2 + k$ where $(h, k)$ is the vertex.
In this problem, the vertex is $(4, 6)$ .
The equation becomes: $y = a(x-4)^2 + 6$ .
Since the parabola opens downwards, $a$ must be negative, implying $a = -1$ .
Thus, the equation is $y = -(x-4)^2 + 6$ .

By comparing our derived expression with the options provided:

$y=-(x-4)^2+6$ matches our derived equation.

Therefore, the correct equation is $y=-(x-4)^2+6$ , corresponding to choice 3.

Final Answer

$y=-(x-4)^2+6$

Key Points to Remember

Essential concepts to master this topic

Vertex Form: Use $y = a(x-h)^2 + k$ where $(h,k)$ is vertex
Technique: Substitute vertex (4,6): $y = a(x-4)^2 + 6$
Check: Downward opening means $a < 0$ , so $a = -1$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative sign for downward parabolas
Don't assume all parabolas open upward = wrong direction! The graph clearly opens downward, so the coefficient must be negative. Always check if the parabola opens up (positive a) or down (negative a).

Practice Quiz

Test your knowledge with interactive questions

Which equation represents the function:

$$ y=x^2 $$

moved 2 spaces to the right

and 5 spaces upwards.

FAQ

Everything you need to know about this question

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

Look at the direction of the curve in the graph! If it opens upward like a U-shape, then $a > 0$ . If it opens downward like an upside-down U, then $a < 0$ .

What if I can't see the vertex clearly on the graph?

The vertex is the highest or lowest point of the parabola. In this problem, it's labeled as (4,6), which becomes your $h$ and $k$ values.

Why is the vertex form better than standard form here?

Vertex form $y = a(x-h)^2 + k$ directly shows you the vertex coordinates and makes it easy to write the equation when you know the vertex.

How do I find the value of 'a' if it's not given?

Use another point on the parabola! Substitute the coordinates into your equation and solve for $a$ . In this case, the graph's direction tells us $a = -1$ .

What's the difference between (x-4) and (x+4) in the equation?

The sign is opposite of the x-coordinate! If the vertex is at $x = 4$ , you write (x-4). If it were at $x = -4$ , you'd write (x+4).

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