How do you find the vertex, the y-intercept, and symmetric point, and use these to sketch the graph given #y=5x^2-30x+31#?

Question

Abigail Allen · Accepted Answer

To do this we will need to perform some algebraic manipulation.

Vertex To find the vertex, we need the #x# and #y# location. There is a handy formula (when the equation is in standard form) for the #x#-coordinate: #x = -b/"2a"#. To find the #y#-coordinate, we can plug in the #x# value we got for the vertex into the original equation. Doing this gives #x = -(-30/"2*5") = 3# Plugging in #x# gives #y = 5*3^2 - 30*3 + 31 = -14# Vertex location = #(3,-14)# Y-intercept The #y#-intercept is when the function crosses the #y#-axis. We can think of this as when the value of #x# is equal to #0#. Doing this gives #y = 5*0^2 - 30*0 + 31 = 31#. Y-intercept = #31#. Symmetric Point A quadratic equation such as this will produce a parabola, which is always symmetric about its vertex. This means that the symmetric point will be the vertex. Graphing Mark a point where the vertex is, then mark the point with the #y#-intercept, and use the concept of symmetry to draw a third point at #(6,31)#. Hint: Draw a dashed vertical line over the vertex, and show that this line is a line of symmetry. After you have done this, it is reasonable to sketch the parabola.

Eli Assouline · Answer

undefined To find the vertex of the parabola represented by the equation y = 5x^2 - 30x + 31, you can use the formula for the x-coordinate of the vertex, which is given by x = -b/(2a) where a and b are the coefficients of x^2 and x respectively. Then substitute this x-value into the equation to find the corresponding y-value.

The y-intercept is found by setting x = 0 in the equation and solving for y.

The symmetric point is the point on the parabola that lies on the same horizontal line as the vertex but on the opposite side of the axis of symmetry.

Once you have found the vertex, y-intercept, and symmetric point, you can sketch the graph by plotting these points and noting the shape of the parabola.