Question

1. sketch the graph of a quadratic equation that has no real solutions.

Explanation:

Step1: Recall Quadratic Graph Properties

A quadratic function is \( y = ax^2 + bx + c \). Its graph is a parabola. For no real solutions, the discriminant \( D = b^2 - 4ac < 0 \), meaning the parabola doesn't intersect the x - axis. Also, if \( a>0 \), the parabola opens upward; if \( a < 0 \), it opens downward. Let's choose \( a = 1 \), \( b = 0 \), \( c = 1 \), so the function is \( y=x^{2}+1 \).

Step2: Analyze the Function \( y = x^{2}+1 \)

• Vertex: The vertex of \( y = ax^{2}+bx + c \) is at \( x=-\frac{b}{2a} \). For \( y=x^{2}+1 \), \( a = 1 \), \( b = 0 \), so \( x = 0 \). Substituting \( x = 0 \) into the function, we get \( y=0 + 1=1 \). So the vertex is \( (0,1) \).
• Direction of Opening: Since \( a=1>0 \), the parabola opens upward.
• Intercepts: To find the x - intercepts, set \( y = 0 \), then \( x^{2}+1=0\Rightarrow x^{2}=- 1 \), which has no real solutions (so no x - intercepts). The y - intercept is when \( x = 0 \), \( y = 1 \), so the y - intercept is \( (0,1) \) (which is also the vertex here).

Step3: Sketch the Graph

1. Plot the vertex \( (0,1) \).
2. Since the parabola opens upward and has no x - intercepts, draw a U - shaped curve (parabola) with the vertex at \( (0,1) \), opening upwards, and not crossing the x - axis. We can also plot a few more points: when \( x = 1 \), \( y=1 + 1=2 \); when \( x=-1 \), \( y=(-1)^{2}+1 = 2 \); when \( x = 2 \), \( y=4 + 1=5 \); when \( x=-2 \), \( y=4 + 1=5 \). Then connect these points smoothly to form the parabola.

Answer:

A quadratic function like \( y = x^{2}+1 \) (with \( a = 1 \), \( b = 0 \), \( c = 1 \), discriminant \( D=0 - 4\times1\times1=-4<0 \)) has a parabola opening upward with vertex at \( (0,1) \), no x - intercepts (so no real solutions for \( x^{2}+1 = 0 \)), and the graph is a U - shaped curve (parabola) opening upwards, vertex at \( (0,1) \), passing through points like \( (1,2) \), \( (-1,2) \), etc., and not intersecting the x - axis.