Complete Question 
Find the general expression for the slope of a line tangent to the curve of y=2x^2+4x at the point P(x,y) . Then find the slopes for  and x=0.5. Sketch the curve and the tangent lines. What is the general expression for the slope of a line tangent to the curve of the function y=2x^2+4 at the point P(x,y) ?
 and x=0.5. Sketch the curve and the tangent lines. What is the general expression for the slope of a line tangent to the curve of the function y=2x^2+4 at the point P(x,y) ?
Answer:
The  generally expression for the slope of  is
 is  
The graph is shown on the first uploaded image 
The  generally expression for the slope of  is
 is   
Step-by-step explanation:
From the question we are told that 
   The  equation of the curve is 
First we differentiate the equation 
 So  
      
Therefore the generally expression for the slope tangent to the curve  is
 is   
The  next step is to substitute for x =  3 and  x =  0.5 
 So  for 
     
      
And  for  
       
        
Here m_1  and  m_2 are slops of the curve 
Next we obtain the coordinates of the tangent lines 
So  at 
    
   
So the coordinate for the first tangent line is  
     
At   
       
     
=>  
So the coordinate for the second  tangent line is  
     
Next we obtain the equation for the tangent lines 
  So generally the slope is mathematically represented as 
         
For    and
 and  
        
=>    
For   and
 and  
        
        
Generally the general expression for the slope of a line tangent to the curve of the function y=2x^2+4 at the point P(x,y) is mathematically evaluated by differentiating  y=2x^2+4 as follows 
      