7. \( f(x)=\frac{2}{5} x^{2}-4 x+14 \)

To analyze the function \( f(x) = \frac{2}{5} x^{2} - 4x + 14 \), we can explore various features such as its vertex, intercepts, and whether it has any extrema. Let's start by finding the vertex of the quadratic function, which can be calculated using the formula for the x-coordinate of the vertex: \[ x_v = -\frac{b}{2a} \] where \( a = \frac{2}{5} \) and \( b = -4 \). 1. **Calculate the x-coordinate of the vertex:** \[ x_v = -\frac{-4}{2 \cdot \frac{2}{5}} = \frac{4}{\frac{4}{5}} = 5 \] 2. **Calculate the y-coordinate of the vertex by substituting \( x_v \) back into the function:** \[ f(5) = \frac{2}{5}(5)^{2} - 4(5) + 14 \] Now, let's calculate \( f(5) \). 3. **Calculate \( f(5) \):** \[ f(5) = \frac{2}{5}(25) - 20 + 14 = 10 - 20 + 14 = 4 \] Thus, the vertex of the function is at the point \( (5, 4) \). Next, we can find the y-intercept by evaluating \( f(0) \): 4. **Calculate the y-intercept:** \[ f(0) = \frac{2}{5}(0)^{2} - 4(0) + 14 = 14 \] So, the y-intercept is at the point \( (0, 14) \). Now, let's find the x-intercepts by solving the equation \( f(x) = 0 \): 5. **Set the function equal to zero and solve for \( x \):** \[ \frac{2}{5} x^{2} - 4x + 14 = 0 \] To solve this quadratic equation, we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = \frac{2}{5} \), \( b = -4 \), and \( c = 14 \). Now, let's calculate the discriminant \( b^2 - 4ac \): 6. **Calculate the discriminant:** \[ D = (-4)^{2} - 4 \cdot \frac{2}{5} \cdot 14 \] Now, let's calculate \( D \). 7. **Calculate \( D \):** \[ D = 16 - \frac{112}{5} = 16 - 22.4 = -6.4 \] Since the discriminant is negative, this means there are no real x-intercepts. In summary, the features of the function \( f(x) = \frac{2}{5} x^{2} - 4x + 14 \) are: - Vertex: \( (5, 4) \) - Y-intercept: \( (0, 14) \) - No real x-intercepts (the graph does not cross the x-axis). If you need further analysis or specific calculations, please let me know!