It's a two-component-mix of a Uniform(-1,1) and a Triangular(-1,1).

To see this, let f(x) denote our Uniform(-1,1) pdf:

f = 1/2;  domain[f] = {x, -1, 1};      
PlotDensity[f]

and let g(x) denote our Triangular(-1,1) pdf:

g = (1 + x)/2; domain[g] = {x, -1, 1};      
PlotDensity[g]

Then, define the two-component mix pdf h(x):

h = (1 - α) f + α g  // FullSimplify

which returns output:

$(1 + x \alpha)/2$

with domain of support:

domain[h] = {x, -1, 1}  && {-1 <= α <= 1};  

Here is a plot of the two-component-mix pdf (using mathStatica/Mathematica):

PlotDensity[h /. α -> Range[-5, 5]/5]

Having said all that, I think it deserves its own name (if it does not have one already) and suggest: Linear distribution

It's a two-component-mix of a Uniform(-1,1) and a Triangular(-1,1).

To see this, let f(x) denote our Uniform(-1,1) pdf:

f = 1/2;  domain[f] = {x, -1, 1};      
PlotDensity[f]

and let g(x) denote our Triangular(-1,1) pdf:

g = (1 + x)/2; domain[g] = {x, -1, 1};      
PlotDensity[g]

Then, define the two-component mix pdf h(x):

h = (1 - α) f + α g  // FullSimplify

which returns output:

$(1 + x \alpha)/2$

with domain of support:

domain[h] = {x, -1, 1}  && {-1 <= α <= 1};  

Here is a plot of the two-component-mix pdf (using mathStatica/Mathematica):

PlotDensity[h /. α -> Range[-5, 5]/5]

Having said all that, I think it deserves its own name (if it does not have one already) and suggest: Linear distribution

It's a two-component-mix of a Uniform(-1,1) and a Triangular(-1,1).

To see this, let f(x) denote our Uniform(-1,1) pdf:

f = 1/2;  domain[f] = {x, -1, 1};      PlotDensity[f]

and let g(x) denote our Triangular(-1,1) pdf:

g = (1 + x)/2; domain[g] = {x, -1, 1};      PlotDensity[g]

Then, define the two-component mix pdf h(x):

h = (1 - $\alpha$) f + $\alpha$ g // FullSimplify

which returns output:

(1 + x $\alpha$)/2

with domain of support:

domain[h] = {x, -1, 1}  && {-1 <= α <= 1};  

Here is a plot of the two-component-mix pdf (using mathStatica/Mathematica):

PlotDensity[h /. $\alpha$ -> Range[-5, 5]/5]

Having said all that, I think it deserves its own name (if it does not have one already) and suggest: Linear distribution

It's a two-component-mix of a Uniform(-1,1) and a Triangular(-1,1).

To see this, let f(x) denote our Uniform(-1,1) pdf:

f = 1/2;  domain[f] = {x, -1, 1};      
PlotDensity[f]

and let g(x) denote our Triangular(-1,1) pdf:

g = (1 + x)/2; domain[g] = {x, -1, 1};      
PlotDensity[g]

Then, define the two-component mix pdf h(x):

h = (1 - α) f + α g  // FullSimplify

which returns output:

$(1 + x \alpha)/2$

with domain of support:

domain[h] = {x, -1, 1}  && {-1 <= α <= 1};  

Here is a plot of the two-component-mix pdf (using mathStatica/Mathematica):

PlotDensity[h /. α -> Range[-5, 5]/5]

Having said all that, I think it deserves its own name (if it does not have one already) and suggest: Linear distribution