Problem 66 - Algebra 1: Concepts and Skills

Question

Sketch the graph of the function. $$y=-3 x^{2}-x+7$$

Step-by-Step Solution

Answer

The graph of the function \( y=-3x^2-x+7 \) is a parabola that opens downwards with its vertex at (1/6 , 6.833) and axis of symmetry \( x = 1/6 \).

Step 1: Identify the Coefficients

The quadratic function equation given is \( y = -3x^2 - x + 7 \). So, the coefficient of \( x^2 \) which is \( a \) is -3, the coefficient of \( x \) i.e. \( b \) is -1, and the constant term \( c \) is 7.

Step 2: Find the Axis of Symmetry

The axis of symmetry can be found using the formula \( x = -b / (2a) \). Substituting our \( a \) and \( b \) values into this formula gives us \( x = 1/6 \).

Step 3: Compute the Vertex

The vertex of the function is the point on the graph where the axis of symmetry intersects the parabola. It can be calculated using the formula \( (h, k) \) where \( h \) is the axis of symmetry and \( k \) is the value of y at \( h \). So, \( h = 1/6 \) and \( k = -(3(1/6)^2) - (1/6) + 7 = 6.833 \).

Step 4: Plot the Vertex and Sketch Parabola

Plot the vertex point (1/6 , 6.833) on the grid. Recognize the parabola opens downwards because the coefficient of \( x^2 \) is negative. Draw the parabola remembering that its branches are mirror images of each other across the line of symmetry \( x = 1/6 \).