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Introduction to Quadratic Functions

Quadratic functions are a fundamental concept in mathematics, and understanding their graphs is crucial for various applications in science, engineering, and other fields. In this article, we will explore the graph of a given quadratic function, $h(x) = (x+1)^2 - 4$, and identify its key features, including the $x$-intercept(s), $y$-intercept, vertex, and axis of symmetry.

The Graph of a Quadratic Function

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve that opens upwards or downwards.

The Given Function

The given function is $h(x) = (x+1)^2 - 4$. To understand the graph of this function, we need to expand the squared term and simplify the expression.

Expanding the Squared Term

Using the formula $(a+b)^2 = a^2 + 2ab + b^2$, we can expand the squared term in the given function:

$h(x) = (x+1)^2 - 4$

$h(x) = x^2 + 2x(1) + 1^2 - 4$

$h(x) = x^2 + 2x + 1 - 4$

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

Identifying the Key Features of the Graph

Now that we have simplified the expression, we can identify the key features of the graph of the given function.

x-Intercept(s)

The $x$-intercept(s) of a graph are the points where the graph intersects the $x$-axis. To find the $x$-intercept(s) of the given function, we need to set $y = 0$ and solve for $x$.

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

$0 = x^2 + 2x - 3$

$x^2 + 2x - 3 = 0$

$(x+3)(x-1) = 0$

$x+3 = 0$ or $x-1 = 0$

$x = -3$ or $x = 1$

Therefore, the $x$-intercept(s) of the graph of the given function are $(-3, 0)$ and $(1, 0)$.

y-Intercept

The $y$-intercept of a graph is the point where the graph intersects the $y$-axis. To find the $y$-intercept of the given function, we need to set $x = 0$ and solve for $y$.

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

$h(0) = 0^2 + 2(0) - 3$

$h(0) = -3$

Therefore, the $y$-intercept of the graph of the given function is $(0, -3)$.

Vertex

The vertex of a parabola is the point where the parabola changes direction. To find the vertex of the graph of the given function, we need to use the formula $x = -\frac{b}{2a}$.

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

$a = 1$ and $b = 2$

$x = -\frac{2}{2(1)}$

$x = -1$

Now that we have found the $x$-coordinate of the vertex, we can find the $y$-coordinate by substituting $x = -1$ into the function.

$h(-1) = (-1)^2 + 2(-1) - 3$

$h(-1) = 1 - 2 - 3$

$h(-1) = -4$

Therefore, the vertex of the graph of the given function is $(-1, -4)$.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. To find the equation of the axis of symmetry, we need to use the formula $x = -\frac{b}{2a}$.

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

$a = 1$ and $b = 2$

$x = -\frac{2}{2(1)}$

$x = -1$

Therefore, the equation of the axis of symmetry is $x = -1$.

Conclusion

In this article, we have explored the graph of a given quadratic function, $h(x) = (x+1)^2 - 4$, and identified its key features, including the $x$-intercept(s), $y$-intercept, vertex, and axis of symmetry. We have used various mathematical techniques, including expanding the squared term, solving quadratic equations, and using the formula for the vertex and axis of symmetry. By understanding the graph of a quadratic function, we can gain a deeper appreciation for the beauty and complexity of mathematics.

Introduction

In our previous article, we explored the graph of a given quadratic function, $h(x) = (x+1)^2 - 4$, and identified its key features, including the $x$-intercept(s), $y$-intercept, vertex, and axis of symmetry. In this article, we will answer some frequently asked questions about quadratic function graphs.

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. A linear function, on the other hand, is a polynomial function of degree one, which means the highest power of the variable is one. Quadratic functions have a parabolic shape, while linear functions have a straight line shape.

Q: How do I find the $x$-intercept(s) of a quadratic function?

A: To find the $x$-intercept(s) of a quadratic function, you need to set $y = 0$ and solve for $x$. This will give you the points where the graph intersects the $x$-axis.

Q: How do I find the $y$-intercept of a quadratic function?

A: To find the $y$-intercept of a quadratic function, you need to set $x = 0$ and solve for $y$. This will give you the point where the graph intersects the $y$-axis.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point where the parabola changes direction. It is the minimum or maximum point of the graph, depending on whether the parabola opens upwards or downwards.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you need to use the formula $x = -\frac{b}{2a}$. This will give you the $x$-coordinate of the vertex. Then, you can substitute this value into the function to find the $y$-coordinate.

Q: What is the axis of symmetry of a quadratic function?

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. It is the line that divides the parabola into two equal parts.

Q: How do I find the equation of the axis of symmetry of a quadratic function?

A: To find the equation of the axis of symmetry of a quadratic function, you need to use the formula $x = -\frac{b}{2a}$. This will give you the equation of the axis of symmetry.

Q: Can a quadratic function have more than one $x$-intercept?

A: Yes, a quadratic function can have more than one $x$-intercept. This occurs when the graph intersects the $x$-axis at two or more points.

Q: Can a quadratic function have a negative $y$-intercept?

A: Yes, a quadratic function can have a negative $y$-intercept. This occurs when the graph intersects the $y$-axis at a point below the $x$-axis.

Q: Can a quadratic function have a vertex that is not on the $x$-axis?

A: Yes, a quadratic function can have a vertex that is not on the $x$-axis. This occurs when the parabola opens upwards or downwards and the vertex is located at a point that is not on the $x$-axis.

Conclusion

In this article, we have answered some frequently asked questions about quadratic function graphs. We have covered topics such as the difference between quadratic and linear functions, finding $x$-intercepts and $y$-intercepts, the vertex and axis of symmetry, and more. By understanding these concepts, you can gain a deeper appreciation for the beauty and complexity of mathematics.