Answer:
Please find attached the graph of the function created with MS Excel showing the relevant points required to draw an approximate graph of the function on a graph paper
Step-by-step explanation:
The given quadratic function is f(x) = -2·x² + 7·x + 4
The range of the input (x) values = -1 ≤ x ≤ 5
The coefficient of the quadratic is negative -2, the graph is n shape
The intercept form of the function is given as follows;
-2·x² + 7·x + 4 = -1 × (2·x² - 7·x - 4)
-1 × (2·x² - 7·x - 4) = -1 × (2·x² + x - 8·x - 4)
-1 × (2·x² + x - 8·x - 4) = -1 × (x · (2·x + 1) - 4·(2·x + 1))
∴ -1 × (x · (2·x + 1) - 4·(2·x + 1)) = -1 × (2·x + 1)·(x - 4)
∴ f(x) = -2·x² + 7·x + 4 = -1 × (2·x + 1)·(x - 4)
At the x-intercepts, (2·x + 1) = 0 or (x - 4) = 0, which gives;
x = -1/2 or x = 4
Therefore, the x-intercepts are (-1/2, 0), (4, 0)
The equation in vertex form is given as follows;
f(x) = -2·x² + 7·x + 4 = -2·(x² - 7·x/2 + 2)
By applying completing the squares method, to x² - 7·x/2 - 2, we get;
Where x² - 7·x/2 - 2
x² - 7·x/2 = 2
x² - 7·x/2 + (-7/4)² = 2 + (-7/4)² = 81/15
(x - 7/4)² = 81/16
∴ (x - 7/4)² - 81/16 = 0 = x² - 7·x/2 - 2
∴ x² - 7·x/2 - 2 = (x - 7/4)² - 81/16
-2·(x² - 7·x/2 + 2) = -2·((x - 7/4)² - 81/16) = -2·(x - 7/4)² + 81/8
The vertex = (7/4, 81/8)
When x = 0, we get;
f(0) = -2 × 0² + 7 × 0 + 4 = 4
The y-intercept = (0, 4)
The sketch of the function should pass through the x-intercepts (-1/2, 0), (4, 0), the y-intercept (0, 4), and the y-intercept (0, 4), and the vertex, (7/4, 81/8) on a graph sheet
Please find attached a drawing of the function of the function created with MS Excel
