Frequently Asked Questions About Parabolas and Quadratic Equations

Oct 13, 2025 by windailys.com 27 views

Which Equation Has a Graph That Is a Parabola with a Vertex at (5,3)?

A parabola is a type of quadratic equation that has a U-shaped graph. The vertex of a parabola is the point at which the graph changes direction, and it is typically represented by the equation (h, k), where h is the x-coordinate and k is the y-coordinate. In this case, we are looking for an equation that has a graph that is a parabola with a vertex at (5,3).

Understanding the Vertex Form of a Parabola

The vertex form of a parabola is given by the equation y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. In this equation, a is the coefficient that determines the direction and width of the parabola. If a is positive, the parabola opens upward, and if a is negative, the parabola opens downward.

Analyzing the Options

Let's analyze each of the options given to determine which one has a graph that is a parabola with a vertex at (5,3).

Option A: y = (x - 5)^2 + 3

This equation is in the vertex form of a parabola, with a vertex at (5,3). The equation is y = (x - 5)^2 + 3, which means that the vertex is at (5,3). The coefficient a is not specified, but it can be any real number. Since the vertex is at (5,3), this equation satisfies the condition.

Option B: y = (x + 5)^2 + 3

This equation is also in the vertex form of a parabola, but the vertex is at (-5,3). The equation is y = (x + 5)^2 + 3, which means that the vertex is at (-5,3). This does not satisfy the condition, as the vertex is not at (5,3).

Option C: y = (x - 3)^2 + 5

This equation is in the vertex form of a parabola, but the vertex is at (3,5). The equation is y = (x - 3)^2 + 5, which means that the vertex is at (3,5). This does not satisfy the condition, as the vertex is not at (5,3).

Option D: y = (x + 3)^2 + 5

This equation is in the vertex form of a parabola, but the vertex is at (-3,5). The equation is y = (x + 3)^2 + 5, which means that the vertex is at (-3,5). This does not satisfy the condition, as the vertex is not at (5,3).

Conclusion

Based on the analysis of each option, the equation that has a graph that is a parabola with a vertex at (5,3) is Option A: y = (x - 5)^2 + 3. This equation is in the vertex form of a parabola, with a vertex at (5,3), and satisfies the condition.

Key Takeaways

The vertex form of a parabola is given by the equation y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
The coefficient a determines the direction and width of the parabola.
The vertex of a parabola is the point at which the graph changes direction.
The equation y = (x - 5)^2 + 3 has a graph that is a parabola with a vertex at (5,3).

Additional Resources

For more information on parabolas and quadratic equations, see the following resources:

Khan Academy: Quadratic Equations
Math Is Fun: Parabolas
Wolfram MathWorld: Parabola

Practice Problems

Try the following practice problems to test your understanding of parabolas and quadratic equations:

Find the vertex of the parabola given by the equation y = (x - 2)^2 + 4.
Determine the direction and width of the parabola given by the equation y = -2(x - 3)^2 + 2.
Find the equation of the parabola with a vertex at (1,2) and a coefficient of 3.
Frequently Asked Questions About Parabolas and Quadratic Equations

In this article, we will answer some of the most frequently asked questions about parabolas and quadratic equations.

Q: What is a parabola?

A: A parabola is a type of quadratic equation that has a U-shaped graph. It is a quadratic function that can be written in the form y = ax^2 + bx + c, where a, b, and c are constants.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point at which the graph changes direction. It is typically represented by the equation (h, k), where h is the x-coordinate and k is the y-coordinate.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula h = -b/2a, where a and b are the coefficients of the quadratic equation. Once you have found the x-coordinate of the vertex, you can plug it into the equation to find the y-coordinate.

Q: What is the standard form of a quadratic equation?

A: The standard form of a quadratic equation is y = ax^2 + bx + c, where a, b, and c are constants. This is the most common form of a quadratic equation and is used to graph the parabola.

Q: What is the vertex form of a quadratic equation?

A: The vertex form of a quadratic equation is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form is used to graph the parabola and is often used in algebra and calculus.

Q: How do I graph a parabola?

A: To graph a parabola, you can use the standard form of the quadratic equation, y = ax^2 + bx + c. You can also use the vertex form of the equation, y = a(x - h)^2 + k, to graph the parabola.

Q: What is the axis of symmetry of a parabola?

A: The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. It is used to graph the parabola and is often used in algebra and calculus.

Q: How do I find the axis of symmetry of a parabola?

A: To find the axis of symmetry of a parabola, you can use the formula x = -b/2a, where a and b are the coefficients of the quadratic equation. This will give you the x-coordinate of the axis of symmetry.

Q: What is the x-intercept of a parabola?

A: The x-intercept of a parabola is the point at which the graph crosses the x-axis. It is typically represented by the equation (x, 0), where x is the x-coordinate.

Q: How do I find the x-intercept of a parabola?

A: To find the x-intercept of a parabola, you can set the equation equal to zero and solve for x. This will give you the x-coordinate of the x-intercept.

Q: What is the y-intercept of a parabola?

A: The y-intercept of a parabola is the point at which the graph crosses the y-axis. It is typically represented by the equation (0, y), where y is the y-coordinate.

Q: How do I find the y-intercept of a parabola?

A: To find the y-intercept of a parabola, you can plug x = 0 into the equation and solve for y. This will give you the y-coordinate of the y-intercept.