Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of

Write the quadratic function in standard form and sketch its...

Key Concepts

Quadratic Function

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. It produces a parabolic graph that opens upward if a is positive and downward if a is negative. This function is foundational in algebra and is crucial for understanding concepts like optimization and the behavior of parabolic curves.

Standard Form

The standard form of a quadratic function is expressed as f(x) = ax² + bx + c. This form is vital because it clearly reveals the coefficients that determine the parabola's shape and position, and it serves as a starting point for methods such as completing the square and applying the quadratic formula.

Completing the Square

Completing the square is a method used to convert the standard form of a quadratic function into the vertex form. This technique involves rearranging and factoring the quadratic expression to identify the vertex of the parabola, which in turn simplifies the process of graphing the function.

Vertex

The vertex of a quadratic function is the point where the graph attains its maximum or minimum value. It is a critical feature of the parabola, as it determines the turning point of the curve. Locating the vertex allows for a more accurate graph sketch and provides insights into the function’s optimal value.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, effectively dividing the graph into two mirror-image halves. This concept is important because it not only aids in sketching the graph but also emphasizes the inherent symmetry present in quadratic functions.

x-Intercepts

The x-intercepts, or roots, of a quadratic function are the points where the graph crosses the x-axis. These intercepts can be found by solving the equation f(x) = 0, typically using factoring, the quadratic formula, or completing the square. They provide essential information about the function's behavior and its intersections with the coordinate axes.

Graphing Parabolic Functions

Graphing a quadratic function involves identifying key features such as the vertex, axis of symmetry, and intercepts. By plotting these features and understanding the direction in which the parabola opens (determined by the coefficient a), one can sketch an accurate representation of the quadratic function, which is a fundamental skill in algebra and precalculus.

Transcript

00:01 Okay, here we have another question that asks us to take the function f of x and get it into standard form, like m of x here.

00:09 Now, the first thing i notice is that we have a coefficient in front of our x term.

00:14 So i'm going to divide everything in this equation by four.

00:20 Or rather, i'm going to factor out of four from this side of the equation.

00:23 So we're going to have four times the quality of x squared minus x plus 21 over four.

00:30 And notice when we distribute the 4 to the 21 force, the 4 is canceled, and we just end up with the 21 back.

00:38 So that's what we have for f of x now.

00:42 Now this, x squared minus x, those are the first two terms of x minus 1⁄2.

00:51 When x minus 1⁄2 quantity squared when it's expanded out.

00:55 And so the last term of that sequence when expanded is going to be 1 .5.

01:01 4.

01:02 So we just need to subtract 20 over 4 from this, which when multiplied by this 4 is the same thing as subtracting 20 from both sides.

01:13 So just remember that because i'm going to subtract 20 from this side instead of doing the 24, 7, having to multiply everything by 4 again.

01:22 So what we have is f of x minus 20 is equal to 4 times the quantity of x minus 1 .5.

01:33 Now i can just move that 20 over x equals 4 times x minus 1 1 1⁄2 squared plus 20.

01:41 And now we have our equation in standard form.

01:44 The good thing about that is that our vertex is really easy to find.

01:47 It's just 1 .520 because it's of the form hk.

01:54 Okay, and so now our axis of symmetry is going to be x equal to 1 half.

02:05 And solving for intercepts, we can do that quickly.

02:09 So that's, i'm going to take this equation and plug in zero for f of x, because that's, all of the solutions lie on this line here, where the y value is zero.

02:20 So i'm going to take 4 times x minus 1⁄2 squared plus 20.

02:28 Now i'm going to subtract 20 from both sides...

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