Question

SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. $$ y=x^{2}+x+4 $$

Short answer

The graph of the function \(y = x^{2}+x+4\) is a parabola opening upwards with the vertex at point (-0.5, 3.75).

Step-by-step solution

Understand the structure of the function

Firstly, understand the quadratic function given. A quadratic function is usually in the form \(ax^{2}+bx+c\), where a, b, and c are constants. In the given quadratic function \(y = x^{2}+x+4\), comparison reveals that \(a = 1\), \(b = 1\), and \(c = 4\).

Find the vertex of the quadratic function

The vertex of a quadratic function in the form \(y = ax^{2} + bx + c\) is given by \((-\frac{b}{2a}, f(-\frac{b}{2a}))\). So for the given quadratic function, the vertex will be at \((-\frac{1}{2(1)}, (1)^{2} - 1/2(1) + 4)\). That will be \((-0.5, 3.75)\).

Plot the graph and the vertex

Now, plot the function on graph paper. Also, ensure to identify and label the vertex \((-0.5, 3.75)\), which is a particular point of interest in the graph of a quadratic function.

Key concepts

Quadratic Function

A quadratic function is a type of polynomial that can be written in the form \( y = ax^2 + bx + c \) where \( a \) , \( b \) , and \( c \) are constants, and \( a \) is not equal to zero. These equations produce a parabolic graph which opens either upwards or downwards depending on the sign of the leading coefficient \( a \). The simplest and most commonly recognized quadratic function is \( y = x^2 \) , where the graph opens upwards and is symmetric around the y-axis.

The behavior of the graph is also influenced by the constants \( b \) and \( c \) , with \( b \) affecting the horizontal position and direction of the parabola, and \( c \) representing the y-intercept, where the graph crosses the y-axis. To understand a quadratic function, one must analyze these coefficients and consider their influence on the graph's shape and position.

Vertex of a Quadratic Function

The vertex of a quadratic function is a significant point on the graph of the parabola, representing the highest or lowest point, depending on the direction in which the parabola opens. It serves as the turning point where the graph changes direction. You can find the vertex by using the formula \( (-\frac{b}{2a}, f(-\frac{b}{2a})) \).

For the given function \( y = x^2 + x + 4 \), by applying the formula, the vertex is located at \( (-\frac{1}{2(1)}, f(-\frac{1}{2(1)})) \) or \( (-0.5, 3.75) \). Knowing the vertex is essential as it helps to provide a clear picture of where to start plotting the graph and ensures an accurate representation of the function's behavior.

Plotting Quadratic Functions

When plotting quadratic functions, it is crucial to start by marking the vertex on the graph. With our example \( y = x^2 + x + 4 \), the vertex would be at the coordinate \( (-0.5, 3.75) \).

After plotting the vertex, you can find additional points by selecting x-values and computing their corresponding y-values. Then, plot these points on the graph to begin shaping the curve of the parabola. The symmetry of a quadratic graph allows for inferring points on the opposite side of the vertex horizontally. Finally, draw a smooth curve through all the points plotted, including the vertex, ensuring that the curve you sketch mirrors correctly across the vertex's line of symmetry.

Quadratic Function Standard Form

The quadratic function standard form is written as \( y = ax^2 + bx + c \), where \( a \) , \( b \) , and \( c \) are known values, and \( x \) and \( y \) are variables. This form provides a straightforward method for identifying the quadratic's characteristics, such as the direction of opening, width of the parabola, and the initial y-intercept at \( c \).

In the equation \( y = x^2 + x + 4 \), this is the standard form, and by looking at the coefficients, you can tell that the parabola opens upwards because \( a = 1 \) is positive. The y-intercept is at \( y = 4 \) when \( x = 0 \). Mastering this standard form is essential as it makes it easier to convert to other forms like vertex or factored form, which may be more useful for certain applications or in finding other characteristics of the quadratic function.