A quadratic function, represented by the formula, y = a(x-h)2 + k, produces a parabola or a “U-shaped” function. Based off of the equation, the value of a determines the “steepness” of the parabola. In addition, the value of a also tells us whether or not the curve opens downwards or upwards. For example, if a is negative, the parabola will open downwards, while if a is positive, the parabola will open upwards. In addition, the variable h in the equation is the horizontal (x-axis) value, and the variable k is the vertical (y-axis) value. In the function equation, as h increase, the parabola shifts to the right. For example, if h=2, that means the parabola is shifted to the right 2 units along the x-axis. On the other hand, as k increases, the parabola shifts upwards along the y-axis. If the k value was negative, however, the parabola will shift downwards along the y-axis. For example, if k=-4, the parabola will shift upwards 4 units along the y-axis.
An exponential function, represented by the equation, y = ab(x – h) +k, produces a line that experiences a consistent growth. In other words, the value of the mathematical function is proportional to the function’s current value. According to the equation, the value of a determines a couple things. For instance, if a is positive, the graph faces upwards, whereas if a is negative, the graph reflects across the x-axis and faces downwards. In addition, the value of b is generally known as the base of the exponent. If b is greater than 1, it is known as the “growth factor,” and if b is less than 1, it is known as the “decay factor.” The value of b must always stay positive. Also, in the exponential function, h represents the horizontal shift “h”-units. When h is positive, the function shifts to the right, whereas if the value of h is negative, the function shifts to the left. Finally, the value of k in the exponential function vertically affects the graph when k is positive and negative. If the value of k is positive, the graph will shift upwards “k”-units, and if the value of k is negative, the graph will shift downwards “k”-units along the y-axis.
A radical function, represented by the equation, y = a√x + h + k, produces a curved line. In this radical equation, the changes in variables a, h, and k are similar to the effects of changing parameters in the other functions mentioned. For example, the value of a results in a vertical stretch of the graph by a factor of a. In addition, the value of h determines the horizontal translation. If h is positive, the graph is shifted to the right “h”-units. However, if h is negative, the graph is shifted to the left “h”-units. Finally, the value of k, similar to the quadratic and exponential equations, determines whether or not the graph shifts upwards or downwards along the y-axis. For instance, if k=10, the graph will shift upwards 10 units along the y-axis.
Last but not least, a rational function, represented by the formula, y = a/(x-h)+ k, produces a hyperbola. According to the rational equation, the value of a affects the graph by increasing or decreasing the curve of the branches of the hyperbola. In other words, if a is a small value, the branches of the hyperbola will come very close to the asymptote. However, if a is a large value, it will be farther away from the asymptotes. Also, the variable h determines where the vertical asymptote will be. If h is positive, the vertical asymptote will shift to the right, whereas if the value of h is negative, the vertical asymptote will shift to the left. Finally, the variable k determines where the horizontal asymptote will be placed. For instance, if the value of k is negative, the asymptote will shift downwards. However, if the value of k is positive, the asymptote will shift upwards.
