In Exercises 13–16, use the vertex and intercepts to sketch the graph of each quadratic function

step 1 Okay, we want to sketch this graph and find its range and domain. So first of all we're going to

In Exercises 13–16, use the vertex and intercepts to sketch...

Key Concepts

Domain and Range

The domain of a quadratic function is all real numbers, as there are no restrictions on the input values x. The range depends on the direction in which the parabola opens: for a parabola opening upward, the range is from the minimum value at the vertex to infinity; for one opening downward, the range is from negative infinity to the maximum value at the vertex.

Intercepts

Intercepts are the points where the graph of a function crosses the coordinate axes. The y-intercept is found by evaluating the function at x = 0, while the x-intercepts (if they exist) are determined by solving f(x) = 0. These points provide important markers that help in sketching an accurate graph of the quadratic 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. In the vertex form f(x) = a(x - h)² + k, the axis of symmetry is given by the equation x = h. This concept is crucial for graphing and analyzing quadratic functions.

Vertex

The vertex of a parabola is the point where the function reaches its maximum or minimum value. It is the turning point of the graph, and its coordinates can be directly obtained from the vertex form. The vertex is essential for understanding the overall behavior of the quadratic function.

Vertex Form

The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form is particularly useful because it immediately reveals the vertex, making it easier to graph the function and determine properties like the axis of symmetry and the direction in which the parabola opens.

Quadratic Function

A quadratic function is a second-degree polynomial, generally expressed as ax² + bx + c, whose graph is a parabola. This function is central in algebra and calculus and models many types of phenomena. Its properties, such as symmetry and the presence of a vertex, make it a key subject in the study of functions.

Transcript

00:01 Okay, we want to sketch this graph and find its range and domain.

00:05 So first of all we're going to do here to get all our information is just going to expand this.

00:10 And i hope you know that x plus 1 squared is just going to be x squared plus 2x plus 1 and we need to minus that.

00:16 So putting that here, we've got minus x squared minus 2x plus 3 because we love 1 times 1, square, which is 1, minus 1, 4 minus 1 is 3.

00:29 Okay, so shouldn't have all the more information we need here.

00:34 Okay, so first off, let's try and find the y intersect.

00:37 So the y intercept when x is zero.

00:39 So i've got zero term here, zero term here.

00:41 Well, it's going to intercept the y axis at three.

00:47 Next up, it's going to intercept the y axis when x plus one squared is equal to four.

00:56 I hope you can see that because intercepts the x -axis.

00:59 X x x x x x x x x x squared to 0 well this thing this x plus 1 square would use even this 4 because got the minus in front so there's two different ways about to go about this obviously so x plus 1 square is equal to 4 so you can simply do this by making this equal to 0 bring it across but might easier this way square root both sides x plus 1 now this is a crucial bit is equal to plus or minus 2 because when you square root minus 2 times minus 2 is actually positive 4.

01:33 So take one off both sides.

01:35 You should have x is equal to minus 3 or x is equal to 1.

01:40 So we know it's going to intercept around here, minus 3 and some around here when x is 1.

01:50 Okay? now our third picture in front of the x squared is going to be negative, so it's going to look something like this...

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