Graph the quadratic function. Specify the vertex, axis of symmetry, maximum or minimum value, an

step 1 We have y is equal to x squared minus 2x minus 3. In order to do this one, we're going to do two

Graph the quadratic function. Specify the vertex, axis of...

Key Concepts

Maximum/Minimum Value

The maximum or minimum value of a quadratic function is found at the vertex. This value is the highest point if the parabola opens downward or the lowest point if it opens upward. It is essential for understanding the range and overall behavior of the function.

Intercepts

Intercepts are the points where the graph of the quadratic function crosses the coordinate axes. The y-intercept is found by evaluating the function at x = 0, while the x-intercepts (or roots) are the solutions to the equation ax² + bx + c = 0. These intercepts are critical points that help in sketching the overall graph of the function.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. Its equation is given by x = -b/(2a) for a quadratic function in the form y = ax² + bx + c, and it is an important feature for analyzing the symmetry in the graph.

Parabola

The graph of a quadratic function is a parabola, a symmetric curve with a distinct vertex. The symmetry of a parabola makes it easier to analyze its properties and graph it accurately. The shape and position of the parabola depend on the coefficients of the quadratic function.

Quadratic Function

A quadratic function is a second-degree polynomial function of the form y = ax² + bx + c. Its graph is a parabola, and the sign of the coefficient a determines whether the parabola opens upward (a > 0) or downward (a < 0). This type of function is widely used to model various real-world phenomena and has unique properties that make it predictable and graphically notable.

Vertex

The vertex of a quadratic function is the highest or lowest point on the parabola, depending on its orientation. For an upward opening parabola, the vertex represents the minimum point, whereas for a downward opening parabola, it represents the maximum point. The vertex can be found using formulas or by completing the square.

Transcript

00:01 We have y is equal to x squared minus 2x minus 3.

00:06 In order to do this one, we're going to do two things.

00:11 First thing is we're going to split this up into point vertex form.

00:16 So we'll complete the square, meaning we have y is equal to x squared minus 2x, minus 1, or plus 1, excuse me, minus 1, minus 3.

00:31 And notice both of these ones here cancel out.

00:37 We have the same equation as we would without them.

00:42 So, y is going to be equal to x minus 1 squared now, with the plus 1, and then we have minus 4, with minus 1 and minus 3.

00:54 So now we can see that we have our vertex at x is equal to 1, and y is equal to negative 4.

01:06 So aside from that, we can take our equation, y is equal to x squared minus 2x minus 3 and we can find the x intercepts of this meaning we factor out two terms so what multiplies to negative 3 and adds up to negative 2 negative 3 and a positive 1 so once that equal to 0 we get x minus 3 is equal to 0 x plus 1 is equal to 0 x plus 1 is equal to 0, meaning then we have x equal to 3 and x equal to negative 1.

02:09 So now we have our vertex and we have our x intercepts.

02:14 So make it a lot easier to graph now.

02:41 Okay.

02:43 So now we can start graphing.

02:46 I'm just going to write our equation at the bottom right here.

02:49 It's going to be y is equal to x squared minus 2x minus 3.

02:55 Now if we plug in 0, we'll get 0 squared minus 2 times.

02:59 0 minus 3.

03:01 So we've been point at negative 3.

03:05 We found from our point vertex form that our vertex is going to be at 1 negative 4.

03:11 Right here.

03:14 If we keep going up, we go to the x equals 2.