Question 1332021 Test 5 Graphing Quadratic Fun ✓ Solved

Test #5 is focused on Graphing Quadratic Functions.

Paper For Above Instructions

Understanding quadratic functions and their properties is crucial in algebra. This paper discusses the essential aspects of graphing quadratic functions, including the standard form, vertex form, and intercept form of quadratic equations, as well as the significance of the vertex, axis of symmetry, and the direction of the parabola.

Introduction to Quadratic Functions

A quadratic function can be expressed in the standard form as:

$f(x)\; =\; ax^2\; +\; bx\; +\; c$

where $a$, $b$, and $c$ are constants and $a\; \ne \; 0$. The graph of a quadratic function is a parabola that opens upwards if $a\; >\; 0$ and downwards if .

Graphing Quadratic Functions

To graph a quadratic function, it is essential to identify the following critical features:

• Vertex: The vertex of the parabola represents the maximum or minimum point of the graph. The vertex can be found using the formula $x\; =\; -b/(2a)$ to find the x-coordinate, and substituting that back into the function to find the y-coordinate.
• Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex. It divides the parabola into two mirror-image halves. The equation for the axis of symmetry is $x\; =\; -b/(2a)$.
• X-intercepts: These are the points where the graph crosses the x-axis, found by setting $f(x)\; =\; 0$ and solving the quadratic equation.
• Y-intercept: The y-intercept occurs when $x\; =\; 0$. It can be found directly from the function as $f(0)\; =\; c$.

Forms of Quadratic Functions

Quadratic functions can be represented in different forms:

• Standard Form: $f(x)\; =\; ax^2\; +\; bx\; +\; c$
• Vertex Form: $f(x)\; =\; a(x\; -\; h)^2\; +\; k$, where $(h,\; k)$ is the vertex of the parabola.
• Intercept Form: $f(x)\; =\; a(x\; -\; p)(x\; -\; q)$, where $p$ and $q$ are the x-intercepts.

Example: Graphing a Quadratic Function

Let's consider the function $f(x)\; =\; 2x^2\; -\; 4x\; +\; 1$. First, determine the vertex:

1. Find the x-coordinate of the vertex:

$x\; =\; -(-4)/(2*2)\; =\; 1$

2. Find the y-coordinate:

$f(1)\; =\; 2(1)^2\; -\; 4(1)\; +\; 1\; =\; -1$

So, the vertex is at $(1,\; -1)$.

Next, determine the x-intercepts by solving:

$2x^2\; -\; 4x\; +\; 1\; =\; 0$. Using the quadratic formula:

$x\; =\; [4\; \pm \; \surd ((-4)^2\; -\; 4(2)(1)]\; /\; (2*2)$

which simplifies to $x\; =\; [4\; \pm \; \surd (16\; -\; 8)]\; /\; 4\; =\; [4\; \pm \; \surd 8]\; /\; 4$, giving us intercepts at

$x\; =\; 2\; \pm \; \surd 2$.

Lastly, the y-intercept is:

$f(0)\; =\; 1$.

Conclusion

Graphing quadratic functions is an essential skill in algebra. By understanding how to identify critical features such as the vertex, axis of symmetry, and intercepts, students can accurately represent quadratic functions visually. Additionally, knowing the various forms of quadratic equations aids in simplifying the graphing process.

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