The directrix and focus of a parabola are equidistant from the vertex. The standard equation for a circle with center ( h, k) and radius r isĮxample: Find the equation of a circle with center (3, 2) and radius 4. * In the table the point (h, k) is the vertex of the parabola and the center of the other conics listed. After making all of these changes to the parent function it looks like this: The +3 added to the parent function indicates a vertical shift upwards 3 units. The indicates a horizontal shift to the right 5 units. The negative in front of the function indicates a reflection in the x-axis. Now look at the shifting and reflecting that has been applied to the parent function. Reflectionįirst start this problem by identifying the parent graph of this function. Graphs can be reflected across the x and y-axis. ![]() Shifting graphs can occur in the vertical horizontal direction. Transformations occur when changes are made to the parent graph. The opposite of division is multiplication therefore our inverse function must be multiplication. In the function there is division so the division must be undone. To find the inverse we must undo what is happening in this function. To find the inverse of a function simply undo what is done in the function.Įxample: Find the inverse of the function. Now we can use these values to plug into the average rate of change equation.Īn inverse function occurs when the coordinate pairs are interchanged. This is found by plugging in the x-value into the function. The average rate of change for two points in the slope of the line through these two points.įirst find the y-value for each x-value given. Set both sets of parenthesis equal to zero and solve for x. To find the zeros of a function you can use factoring or the quadratic equation.Įxample: Find the zeros of the function. The zeros of a function are the x-values for which. An odd function is a function that is symmetric with respect to the origin. Īn even function is a function that is symmetric in respect to the y-axis. Therefore, the function is decreasing from and increasing from. The function is then increasing after the vertex. The function is decreasing until it gets to the vertex at. To solve this we should first graph the equation. Įxample: For what intervals is the function increasing, decreasing, or constant. Ī function is constant on an interval if when analyzing two points. Ī function is decreasing on an interval if when analyzing two points. The function does not exist at x = 0 or y = 0.Ī function is increasing on an interval if when analyzing two points. If we graph this equation we get a graph that looks like:Īs we can see from the graph the domain and range are indeed what we found algebraically. Now that we have solved the domain and range algebraically we can verify our answers using a graphical approach. Therefore, there is not a possible way to every achieve y = 0 because 3 divided by anything will never by 0. Once again we must take into consideration that the x is in the denominator. To solve for the range we must consider all of the y values that are possible for the function. This is the only value for x that is not possible so the domain is. Therefore, x = 0 is not a possible value for the function. If x = 0 then the function would be undefined because 0 cannot be in the denominator. ![]() The x is in the denominator which means some values of x will not be possible. To solve for the domain algebraically we need to consider what x values are possible for this function. The range of a function is the set of all possible outputs, or dependent variables, for the function.ĭomain and range can be found algebraically or by graphing.Įxample: Find the domain and range for the function The domain of a function is the set of all possible inputs, or independent variables, for the function. ![]() If there is any point at which the vertical line passes through the graph of the equation more than once, then it is not a function. Imagine moving a vertical line back and forth across the graph (You can use your pencil or the edge of a ruler for this.). You can tell if an equation is a function by performing the vertical line test. To solve this we will substitute x = 2 into our function. To find the function value given the value of the function, plug in the value of the function given for x.Įxample: Let, find the function values of. ![]() Where is the value of the function, y is the dependent variable, and x is the independent variable. A function is relationship between two variables such that each independent variable has only one dependent variable.
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