Locate the Symmetry Point of f(x) = -4x² + 8x + 3

Q: Why is the vertex called a 'symmetry point'?

The vertex is the axis of symmetry for a parabola! If you fold the graph along the vertical line x = 1, both sides match perfectly. That's why we call it the symmetry point.

Q: How do I know if this is a maximum or minimum point?

Look at the coefficient of x²! Since a = -4 , the parabola opens downward, making (1,7) a maximum point - the highest point on the graph.

Q: Can I use completing the square instead?

Yes! Completing the square gives $ f(x) = -4(x-1)² + 7 $, showing the vertex is (1,7). Both methods work, but the vertex formula is usually faster.

Q: What if I get a fraction for the x-coordinate?

That's completely normal! Many parabolas have fractional x-coordinates for their vertex. Just calculate carefully and substitute the exact fraction back to find the y-coordinate.

Q: How can I check my vertex is correct?

Two ways: (1) The vertex should be equidistant from any two points with the same y-value, and (2) substitute x = 1 back into the original function to verify f(1) = 7.

Quadratic Vertex with Formula Method

Given the expression of the quadratic function

Finding the symmetry point of the function

$f(x)=-4x^2+8x+3$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the point of symmetry in the function

00:03 The point of symmetry is the point where if you fold the parabola in half

00:06 The halves will be equal to each other

00:10 Let's examine the function coefficients

00:13 We'll use the formula to calculate the vertex point

00:16 We'll substitute appropriate values according to the given data and solve for X at the point

00:28 This is the X value at the point of symmetry

00:31 Now we'll substitute this X value in the function to find the Y value at the point

00:47 This is the Y value at the point of symmetry

00:50 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

Given the expression of the quadratic function

Finding the symmetry point of the function

$f(x)=-4x^2+8x+3$

Step-by-step solution

To find the symmetry point of the quadratic function $f(x) = -4x^2 + 8x + 3$ , follow these steps:

Step 1: Identify key parameters
The function is of the form $f(x) = ax^2 + bx + c$ , with $a = -4$ , $b = 8$ , and $c = 3$ .
Step 2: Find the x-coordinate of the vertex
Use the formula for the x-coordinate of the vertex: $x = -\frac{b}{2a}$ .
Substitute the values for $b$ and $a$ :
$x = -\frac{8}{2 \times -4} = -\frac{8}{-8} = 1$ .
Step 3: Find the y-coordinate by substituting back into $f(x)$
Calculate $f(1)$ :
$f(1) = -4(1)^2 + 8(1) + 3 = -4 + 8 + 3 = 7$ .
Step 4: State the symmetry point
The symmetry point, or vertex, of the function is $(1, 7)$ .

Therefore, the symmetry point of the quadratic function $f(x) = -4x^2 + 8x + 3$ is $(1, 7)$ .

Final Answer

$(1,7)$

Key Points to Remember

Essential concepts to master this topic

Vertex Formula: Use x = -b/(2a) to find symmetry point
Technique: Calculate f(1) = -4(1)² + 8(1) + 3 = 7
Check: Vertex (1,7) is highest point since a = -4 < 0 ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative sign in vertex formula
Don't use x = b/(2a) instead of x = -b/(2a) = wrong x-coordinate! This misses the negative sign and shifts the vertex to the wrong side. Always remember the formula is x = -b/(2a) with the negative sign.

Practice Quiz

Test your knowledge with interactive questions

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

$$ f(x)=-3x^2+3 $$

FAQ

Everything you need to know about this question

Why is the vertex called a 'symmetry point'?

The vertex is the axis of symmetry for a parabola! If you fold the graph along the vertical line x = 1, both sides match perfectly. That's why we call it the symmetry point.

How do I know if this is a maximum or minimum point?

Look at the coefficient of x²! Since a = -4 < 0, the parabola opens downward, making (1,7) a maximum point - the highest point on the graph.

Can I use completing the square instead?

Yes! Completing the square gives $f(x) = -4(x-1)² + 7$ , showing the vertex is (1,7). Both methods work, but the vertex formula is usually faster.

What if I get a fraction for the x-coordinate?

That's completely normal! Many parabolas have fractional x-coordinates for their vertex. Just calculate carefully and substitute the exact fraction back to find the y-coordinate.

How can I check my vertex is correct?

Two ways: (1) The vertex should be equidistant from any two points with the same y-value, and (2) substitute x = 1 back into the original function to verify f(1) = 7.

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Locate the Symmetry Point of f(x) = -4x² + 8x + 3

Quadratic Vertex with Formula Method

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why is the vertex called a 'symmetry point'?

How do I know if this is a maximum or minimum point?

Can I use completing the square instead?

What if I get a fraction for the x-coordinate?

How can I check my vertex is correct?

🌟 Unlock Your Math Potential

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