Find the vertex of the graph of each quadratic function....

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and graph the function.
$$
f(x)=\frac{1}{2} x^{2}+4 x+\frac{15}{2}
$$

Key Concepts

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing the graph into two mirror-image halves. This line is given by the equation x = -b/(2a) in the standard form of a quadratic function.

Intercepts

Intercepts are points where the graph of the 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 the solutions to the equation ax² + bx + c = 0, which can be obtained through factoring, completing the square, or using the quadratic formula.

Quadratic Function

A quadratic function is a second-degree polynomial typically expressed in the standard form ax² + bx + c. It is one of the fundamental types of functions in algebra and is known for its parabolic graph, which exhibits key characteristics such as symmetry, a vertex, and intercepts with the axes.

Vertex

The vertex of a quadratic function is the highest or lowest point on its graph, depending on whether the parabola opens downward or upward. It represents the maximum or minimum value of the function and can be calculated using the formula -b/(2a) for the x-coordinate, with the corresponding y-coordinate found by substituting this value into the function.

Direction of Opening

The direction in which a parabola opens is determined by the sign of the leading coefficient a in the quadratic function. If a is positive, the parabola opens upward, indicating that the vertex is a minimum point; if a is negative, the parabola opens downward, and the vertex represents a maximum point.

Transcript

00:01 I want to graph the quadratic function f -of -x equals one -half x squared plus 4x plus 15 halves.

00:07 And i'm going to start by plotting my y intercept because my c value is my y intercept.

00:13 And 15 halves is the same as 7 .5.

00:18 So we're going to put a dot at 7 .5 on our y -intercept.

00:24 Next i'm going to find my vertex.

00:26 And to find my vertex, i am going to use the vertex formula.

00:30 And a is my lead coefficient of one -half, and b is the coefficient of the second term, so positive 4.

00:41 And x equals negative b over 2a, which is negative 4 over 1, which is negative 4.

00:52 So the x value of my vertex is negative 4, and i'm just going to write vertex up here for when i'm done.

01:00 And to get the y value, we're going to evaluate f of negative 4.

01:04 So i've got one half times negative four squared plus four times negative four plus 7 .5, which is the same as 15 halves.

01:14 And so negative four squared, 16 times a half is 8 plus 4 times negative 4.

01:22 So minus 16 plus 7 .5, which is going to give me a negative 0 .5.

01:29 So my vertex is the ordered paired negative 4 comma, negative 0 .5 .0 .5 .5.

01:34 0 .5.

01:39 And i'm going to go ahead and plot that...

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