If 𝑎 is a constant greater than zero, which of the following would be the graph of 𝑦 equals 𝑎 multiplied by 𝑥 plus 𝑎 in the 𝑥𝑦-plane?
Let’s firstly consider the equation 𝑦 equals 𝑎 multiplied by 𝑥 plus 𝑎. We can simplify this equation by expanding the brackets or parenthesis. 𝑎 multiplied by 𝑥 is equal to 𝑎𝑥. And 𝑎 multiplied by 𝑎 equals 𝑎 squared. Therefore, the equation can be rewritten as 𝑦 equals 𝑥 plus 𝑎 squared. This equation has been rewritten in the form 𝑦 equals 𝑚𝑥 plus 𝑏. This is the general equation of a straight line, where 𝑚 is the slope or gradient and 𝑏 is the 𝑦-intercept.
We’re told in the question that 𝑎 is greater than zero. This means that our slope or gradient must be positive. Graphs B and C have a negative gradient as the slope downwards from left to right. We can, therefore, rule out these two options. We need to work out which one of the other two graphs has a 𝑦-intercept that is the square of the slope.
Graph A has a 𝑦-intercept of negative one. As the value of our 𝑦-intercept is 𝑎 squared, it must be positive. We can, therefore, say that graph A is also not correct. Graph D has a 𝑦-intercept of four as this is the point where it crosses the 𝑦-axis. This graph has a slope or gradient of two. For every one square we go across in the 𝑥-direction, we move two squares in the 𝑦-direction.
The equation of line D is 𝑦 equals two 𝑥 plus four. By factoring out a two, this is equal to 𝑦 equals two multiplied by 𝑥 plus two. This equation is in the form 𝑦 equals 𝑎 multiplied by 𝑥 plus 𝑎, where the value of 𝑎 is two, which is greater than zero.
We can, therefore, conclude that our correct answer is option D.